Research On High-fidelity NMR Spectrum Reconstruction Algorithm Based On Hypercomplex Numbers

Nuclear Magnetic Resonance(NMR)spectroscopy uses the phenomenon of magnetic resonance to obtain the spectrum of the object.It plays an indispensable role in chemistry,biology and medicine,such as structure detection,metabolite analysis,quantitative analysis,disease diagnosis,etc.However,since the object becomes more and more complex,the onedimensional NMR suffers from peaks overlap,making it difficult to identify and quantitatively analyze.Multidimensional NMR tends to represent more fruitful biochemical information than1 D NMR.Hypercomplex NMR is a basic format for NMR signal acquisition and storage,which can significantly improve resolution.Each dimension of a hypercomplex NMR spectrum contains real and imaginary parts.Therefore,the forms of one-dimensional NMR are complex numbers,and two-dimensional and higher dimensional NMR are all hypercomplex.However,the acquisition time of hypercomplex NMR exhibits an exponential increase with increasing dimensionality and resolution,and the acquisition time from 2D to 4D spectra rises from minutes to tens of days,due to the limitation of indirect dimensional evolution time.One of the major challenges for hypercomplex NMR is the long sampling time.Non-uniform sampling(NUS)was used to acquire part of the experimental data to speed up the data acquisition.However,NUS breaks the Nyquist sampling,which means the amount of collected data is less than the Nyquist requirement.An undersampled spectrum produces artifacts.Therefore,a proper method is needed to reconstruct the hypercomplex spectrum.State-of-the-art signal reconstruction methods include optimization models and deep learning.Optimization models include compressed sensing and low-rank Hankel matrix.The former reconstructs the signal based on the sparsity constraint,but the broad spectral peak does not satisfy the sparsity assumption,resulting in distortion.The latter is based on low-rank Hankel matrix,which can reconstruct completely broad spectral peaks.The existing methods for processing a two-dimensional hypercomplex signal are to divide it into two complex signals,and then perform a complex data reconstruction algorithm correspondingly.However,this method may lead to distortion,because the similar information contained between different components is not effectively utilized.Therefore,it is necessary to propose new methods for hypercomplex signal reconstruction to overcome the existing problems.In this paper,starting from the mathematical characteristic of hypercomplex NMR signals and combining with the existing spectra reconstruction methods,this paper focus on how to achieve high-fidelity reconstruction of nonuniformly hypercomplex signals.The main work includes:(1)Propose a high-fidelity reconstruction low rank method of hypercomplex NMR spectroscopy(Hypercomplex Low Rank,HLR)and derive its numerical algorithm.The reconstruction results show that,compared with the single-component low-rank Hankel matrix reconstruction method,the proposed method is significantly better than the comparison method in the correlation of low-intensity spectra peaks and robustness to noise.(2)By introducing matrix decomposition,a high-fidelity fast hypercomplex NMR spectroscopy reconstruction method is proposed,and its numerical algorithm is deduced.The reconstruction results show that the proposed method can not only reconstruct the spectrum with high fidelity but also reduce the reconstruction time by nearly 80% by parallel computing.