Uniform Penalty inversion of two-dimensional NMR Relaxation data

TL;DR

This paper introduces the 2DUPEN algorithm, extending the UPEN method to two-dimensional NMR data, enabling automatic, locally adaptive regularization for improved inversion of ill-posed inverse Laplace transform problems.

Contribution

The paper develops a novel 2DUPEN algorithm that extends UPEN to 2D NMR data, with an efficient implementation and automatic regularization parameter selection.

Findings

Accurately reconstructs peaks and flat regions in 2D NMR data.

Automatically adapts regularization parameters without prior noise information.

Demonstrates effectiveness on simulated and real datasets.

Abstract

The inversion of two-dimensional NMR data is an ill-posed problem related to the numerical computation of the inverse Laplace transform. In this paper we present the 2DUPEN algorithm that extends the Uniform Penalty (UPEN) algorithm [Borgia, Brown, Fantazzini, {\em Journal of Magnetic Resonance}, 1998] to two-dimensional data. The UPEN algorithm, defined for the inversion of one-dimensional NMR relaxation data, uses Tikhonov-like regularization and optionally non-negativity constraints in order to implement locally adapted regularization. In this paper, we analyze the regularization properties of this approach. Moreover, we extend the one-dimensional UPEN algorithm to the two-dimensional case and present an efficient implementation based on the Newton Projection method. 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.