Two-dimensional Inversion Problems In Lowfield Nmr Technology

Relaxation time measured by NMR(Nuclear Magnetic Resonance)correlates with many material characteristics like viscosity,crystallinity,concentration,and length of polymer chain.Hence NMR relaxometry,the measurement of relaxation time,has been becoming a powerful analyzing approach.The traditional 1D T2 or T1 relaxometry analysis is rapid,non-invasive and non-destructive.It can be implemented by a low cost,compact low-field NMR equipment.However,when 1D relaxometry analysis is used for the complex mixtures,the peaks of different components are prone to overlap together,resulting in the misinterpretation of the spectra.Herein,the 2D relaxometry is proposed to outcome this obstacle of the 1D relaxometry.Original sampling signals from NMR equipment are too intricate for us to understand.These signals can be transformed into understandable spectra by inversion.Comparing to 1D inversion,2D inversion need solve two main difficult problems.First,the data used by 2D inversion is much larger than that used in 1D case.Consequently,the 2D inversion problem,without data compressing,could not be handled properly by ordinary computers.Second,inverse problems are often ill-posed.The ill-posedness of the 2D inversion problem characterized by instability and non-uniqueness is more serious.A small error in sampling data can result in much larger errors in the inversion spectrum.The major contributions to solve the above-mentioned problems in this dissertation include:(1)Data preprocessing:Replace the traditional 2-step phase cycling sampling method with a 4-step one,which enhances the signal to noise ratio from the source and further enhances the accuracy of the inverted spectrum;Perform singular value decomposition with inversion kernels and calculate the ranks of the kernels according to singular values.Then truncate the kernels using the ranks.Combined with Kronecker product,the data can be compressed adaptively,which significantly reduces computation effort;(2)Improvement of existing algorithms: An accurate progressive refinement TSVD based 2D NMR inversion algorithm was proposed to prevent the algorithm from failure to locate the exact truncating position and generation of artificial small peaks.But the possibility of erasing existing peaks was introduced at the same time;Some more appropriate stopping critias were designed to improve the performance of traditional BRD algorithm,which circumvents the problem causing by the strong dependence of the estimation accurancy of noise level and improves the convergence efficiency;(3)A new L1 regularization based 2D inversion algorithm was proposed in this paper: In most application cases,the spectra are sparse and a too complex spectrum is not interpretable and understandable.The L1 regularization gives preference to solutions with a low density of significant data.Consequently,the application of L1 regularization to implement 2D NMR inversion is suitable.A fast iterative algorithm is proposed to solve the L1 regularization problem.The effectiveness,robustness and practical utility of different algorithms were analyzed by simulations and experiments.The results demonstrate that the proposed L1 regula based algorithm can produce spectrum with higher resolution than traditional inversion algorithms.The L1 regularization based 2D NMR inversion algorithm has great application potential.