An Inversion for LF-NMR Signals Processing with BFGS Algorithm

A powerful tool to process and analyze Nuclear Magnetic Resonance (NMR) signals is inversion, of which essence is to solve the Fredholm integral equation of the first kind with non-negative constraints known as an ill-conditioned problem. In this paper, an inversion method is presented based on the regularization method. The proposed objective function can turn the minimization regularization with non-negative constrains into unconstrained maximization, which is piecewise, quadratic and differentiable. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is applied to deal with the inversion problem, and an automatic method is demonstrated to pick up regularization parameter by combining the L-curve method with generalized cross validation (GCV) method. Compared with other methods, the proposed method is capable of inversing both ID and 2D NMR signals data and obtaining reliable NMR spectrum even in the case of a low SNR. The numerical simulations and practical experiments prove the accuracy and efficiency of the inversion method.