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. 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

Iterative algorithms for a non-linear inverse problem in atmospheric lidar

We consider the inverse problem of retrieving aerosol extinction coefficients from Raman lidar measurements. In this problem the unknown and the data are related through the exponential of a linear operator, the unknown is non-negative and the data follow the Poisson distribution. Standard methods work on the log-transformed data and solve the resulting linear inverse problem, but neglect to take into account the noise statistics. In this study we show that proper modelling of the noise distribution can improve substantially the quality of the reconstructed extinction profiles. To achieve this goal, we consider the non-linear inverse problem with non-negativity constraint, and propose two iterative algorithms derived using the Karush-Kuhn-Tucker conditions. We validate the algorithms with synthetic and experimental data. As expected, the proposed algorithms outperform standard methods in terms of sensitivity to noise and reliability of the estimated profile.Comment: 19 pages, 6 figure

Upen2DTool: A Uniform PENalty Matlabtool for inversion of 2DNMR relaxation data

The analysis of Nuclear Magnetic Resonance (NMR) relaxation of H nuclei gives fundamental information about the structure of different types of porous media via the determination of 1-dimensional and 2-dimensional (2D) distributions of the typical NMR parameters (longitudinal T_1 and transverse T_2 relaxation times). Despite the great amount of literature available on this topic, only a few open source software are available, usually after a direct request to the authors. This work aims to stimulate the comparison of different approaches by making our open source inversion Matlab Package (Upen2DTool) available, to encourage its application and extension to the various 2D NMR problems. The solution algorithm applies the Uniform PENalty principle and automatically computes locally adapted regularization parameters and approximate solutions. Moreover a Windows binary code is available for testing for non-Matlab users

A variational approach to Gibbs artifacts removal in MRI

Gibbs ringing is a feature of MR images caused by the finite sampling of the acquisition space (k-space). It manifests itself with ringing patterns around sharp edges which become increasingly significant for low-resolution acquisitions. In this paper, we model the Gibbs artefact removal as a constrained variational problem where the data discrepancy, represented in denoising and convolutive form, is balanced to sparsity-promoting regularization functions such as Total Variation, Total Generalized Variation and L1 norm of the Wavelet transform. The efficacy of such models is evaluated by running a set of numerical experiments both on synthetic data and real acquisitions of brain images. The Total Generalized Variation penalty coupled with the convolutive data discrepancy term yields, in general, the best results both on synthetic and real data

What are, and what are not, Inverse Laplace Transforms

Time-domain NMR, in one and higher dimensionalities, makes routine use of inversion algorithms to generate results called \T2-distributions' or joint distributions in two (or higher) dimensions of other NMR parameters, T1, diffusivity D, pore size a, etc. These are frequently referred to as \Inverse Laplace Transforms' although the standard inversion of the Laplace Transform long-established in many textbooks of mathematical physics does not perform (and cannot perform) the calculation of such distributions. The operations performed in the estimation of a \T2-distribution' are the estimation of solutions to a Fredholm Integral Equation (of the First Kind), a different and more general object whose discretization results in a standard problem in linear algebra, albeit suffering from well-known problems of ill-conditioning and computational limits for large problem sizes. The Fredholm Integral Equation is not restricted to exponential kernels; the same solution algorithms can be used with kernels of completely different form. On the other hand, (true) Inverse Laplace Transforms, treated analytically, can be of real utility in solving the diffusion problems highly relevant in the subject of NMR in porous media

Assessment of Fluid Transport Mechanisms in Shale Gas Reservoirs

The complex interplay between the physical and flow properties of shales was investigated. A methodology was developed to estimate free and bound porosity fractions from NMR-T2 experiments on shales, while a second order flow model was proposed to interpret gas permeability data. Slippage effects appeared to be influenced by characteristic pore lengths, while poroelastic behaviour was linked to compositional data. Potential associations emerged between FFI/BVI, pore sizes, fluid dynamic phenomena, and shale composition

Characterization of first sorption cycle of white Portland cement by 1H NMR

The work carried out is focused on the exploration of processes occurring in cement materials during sorption cycles by using Nuclear Magnetic Resonance (NMR) relaxometry. Long (months) and short (days-weeks) sorption cycles of cement materials were explored. The long cycle consists of around 6 months of drying and re-wetting cement samples of different sizes and water-to-cement (w/c) ratios in a homemade relative humidity (RH) chamber. Short cycles were performed by drying samples of different sizes and w/c ratios in the oven at 60 ˚C and re-wetting underwater. Different NMR techniques, such as one- and two-dimensional relaxometry and solid-signal analyses, were used to study the samples. Firstly, by the interpretation of quasi-continuous distributions of T2 relaxation time, we demonstrated that some reversible and irreversible changes concerning smaller porosity happened during the first sorption cycle. Secondly, using 2D NMR and a new 2D NMR inversion algorithm we showed preliminary results on the cement T1-T2 maps. Data obtained during sorption processes indicated possible water exchange between different pore populations inside the cement samples. Thirdly, the solid structure of cement samples was qualitatively investigated with T1 measurements and, as far as we know, for the first time interpreted with the Pake-Doublet theory. Changes in the solid structure were observed. Precisely variations of the amount of Ettringite during drying/wetting were proposed to take place. Finally, a work on NMR single-sided equipment design for in situ cement investigation was shown. The multi-cubic-blocks magnet structure design was performed using different specific CAD software, and the magnetic fields generated by RF coils of different geometries were investigated using a customized Matlab script. The single-sided NMR instrument equipped with the designed single-sided magnet and coil was built by the ERICA partner company MR Solutions (Abingdon, UK), and the preliminary results resultsated the correctness of the developed design