A Robust Algorithm for 2-D NMR Diffusion–Relaxation Spectra Inversion

Abstract

A two-step iterative shrinkage/thresholding (TIST) algorithm was proposed to solve the objective function of the L1 regularization for the 2-D nuclear magnetic resonance diffusion–relaxation ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D - T_{2}$ </tex-math></inline-formula> ) spectra inversion. Moreover, a simple and effective method combining the S-curve method and the Chi-square method was proposed to select the optimal L1 regularization parameter. We compared the inverted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D - T_{2}$ </tex-math></inline-formula> spectra derived from the TIST algorithm with those from the truncated singular value decomposition (TSVD), Butler–Reeds–Dawson (BRD), original IST (OIST), and IST algorithms using numerical experiments. The experimental results indicated that the inverted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D - T_{2}$ </tex-math></inline-formula> spectra derived from the OIST, IST, and TIST algorithms were affected by noise to a smaller degree, and the robustness was higher for the OIST, IST, and TIST algorithms than for the TSVD and BRD algorithms. In addition, we compared the accuracy and operating time of the TIST, OIST, and IST algorithms for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D - T_{2}$ </tex-math></inline-formula> spectra inversion and concluded that the three algorithms had an equivalent accuracy but the TIST algorithm had a faster convergence rate.