Uniform Penalty inversion of two-dimensional NMR relaxation data

The inversion of two-dimensional NMR data is an ill-posed problem related to the numerical computation of the inverse Laplace transform. The Uniform Penalty (UPEN) algorithm (Borgia et al 1998 J. Magn. Reson. 132 65–77), defined for the inversion of one-dimensional NMR relaxation data, uses Tikhonov-like regularization and optional lower bound constraints in order to implement locally adapted regularization. In this paper, we analyze the reg- ularization properties of this approach. Moreover, we extend the one-dimen- sional UPEN algorithm to the two-dimensional case and present an efficient implementation based on the Newton Projection method (2DUPEN). Without any a-priori information on the noise norm, 2DUPEN automatically computes the locally adapted regularization parameters and the distribution of the unknown NMR parameters by using variable smoothing. Results of numerical experiments on simulated and real data are presented in order to illustrate the potential of the proposed method in reconstructing peaks and flat regions with the same accuracy.