On the inversion of diffusion NMR data: Tikhonov regularization and optimal choice of the regularization parameter

• Published in: Journal of magnetic resonance (1997) Vol. 211; no. 2; pp. 178 - 185
• Main Author: Day, Iain J.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.08.2011
• Subjects: Algorithms; Azo Compounds - chemistry; Coloring Agents - chemistry; Data Interpretation, Statistical; Diffusion; Diffusion NMR; Inverse Laplace transform; Inversion; Inversions; Magnetic Resonance Spectroscopy - methods; Magnetic Resonance Spectroscopy - statistics & numerical data; Mathematical models; Models, Statistical; Molecular Weight; Normal Distribution; Nuclear magnetic resonance; Optimization; Parameter choice; Polymers; Polystyrenes; Regularization; Reproducibility of Results; Tikhonov regularization; Diffusion NMR; Inversion; Parameter choice; Tikhonov regularization; Inverse Laplace transform
• ISSN: 1090-7807; 1096-0856
• DOI: 10.1016/j.jmr.2011.05.014

[Display omitted] ► The inversion of diffusion NMR data has been performed using Tikhonov regularization. ► Automatic, optimal choice of the regularization parameter is demonstrated for the L-curve and GCV methods. ► The technique is computationally straightforward using standard matrix algebra. ► The approach is demonstrated using sunset yellow and poly(styrene 4-sulphonate). The analysis of diffusion NMR data in terms of distributions of diffusion coefficients is hampered by the ill-posed nature of the required inverse Laplace transformation. Naïve approaches such as multiexponential fitting or standard least-squares algorithms are numerically unstable and often fail. This paper updates the CONTIN approach of the application of Tikhonov regularization to stabilise this numerical inversion problem and demonstrates two methods for automatically choosing the optimal value of the regularization parameter. These approaches are computationally efficient and easy to implement using standard matrix algebra techniques. Example analyses are presenting using both synthetic data and experimental results of diffusion NMR studies on the azo-dye sunset yellow and some polymer molecular weight reference standards.