Study on Multi-Exponential Inversion Method for NMR Relaxation Signals with Tikhonov Segularization

The analysis of NMR data in terms of inversion of relaxation distribution is hampered by the ill-posed nature of the required solution about the Fredholm integral equation. Naive approaches such as multi-exponential fitting or standard least-squares algorithms are numerically unstable and often failed. This paper updates the application of Tikhonov regularization to stabilize this numerical inversion problem and demonstrates the method for automatically choosing the optimal value of the regularization parameter. The approach is computationally efficient and easy to implement using standard matrix algebra techniques, which is based on mathematical ware MATLAB. Example analyses are presented using both synthetic data and experimental results of NMR studies on the liquid samples like as oils and yoghurt.