Uniform Penalty inversion of two-dimensional NMR relaxation data

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, 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.

L1, Lp, L2, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

“…26 Basser et al 8 implemented L 1 -penalized relaxometry in the context of the development of a marginal distribution constrained optimization (MADCO) method for constrained reconstruction of 2D relaxometry distributions. Samples were obtained from an arthroplasty procedure.…”

“…This may be attributed to tissue breakdown within the osteoarthritic cartilage relaxometry. 26 Basser et al 8 implemented L 1 -penalized relaxometry in the context of the development of a marginal distribution constrained optimization (MADCO) method for constrained reconstruction of 2D relaxometry distributions. While commenting that this penalty is likely to be more applicable to the sparse solution matrices obtained in a typical experiment, comparison of penalties was not performed.…”

Multidimensional Proton Nuclear Magnetic Resonance Relaxation Morphological and Chemical Spectrum Graphics for Monitoring and Characterization of Polyunsaturated Fatty‐Acid Oxidation

“…To maximize the amount of information that can be extracted from LF‐NMR signal generation, we adapted and used a PDCO solver with excellent reconstruction accuracy that outperformed other in‐house NMR instrument's mathematical reconstruction programs (Wiesman et al, ). Signal reconstruction is essentially an ill‐posed ILT problem (Bortolotti et al, ; Reci et al, ); this implies that similar relaxation curves and noise in the measurements can result in very different reconstruction spectra and/or a high signal‐to‐noise ratio (SNR). In previous reports (Berman et al, ; Campisi‐Pinto et al, ; Wiesman et al, ), it was demonstrated that if an l 2 regularization term is added to the objective function of a common regularization, as seen in Eq.…”

1H LF‐NMR Energy Relaxation Time Characterization of the Chemical and Morphological Structure of PUFA‐Rich Linseed Oil During Oxidation With and Without Antioxidants

“…NMR signal reconstruction is an ill‐posed inverse Laplace transform (ILT) problem . Therefore, very similar energy relaxation time curves and/or a relatively low noise level in the measurements can result in very different reconstructed spectra and/or signal to noise ratio (SNR).…”