A crucial issue in two-dimensional Nuclear Magnetic Resonance (NMR) is the speed and accuracy of the data inversion. This paper proposes a multi-penalty method with locally adapted regularization parameters for fast and accurate inversion of 2DNMR data. The method solves an unconstrained optimization problem whose objective contains a data-fitting term, a single $L1$ penalty parameter and a multiple parameter $L2$ penalty. We propose an adaptation of the Fast Iterative Shrinkage and Thresholding (FISTA) method to solve the multi-penalty minimization problem, and an automatic procedure to compute all the penalty parameters. This procedure generalizes the Uniform Penalty principle introduced in [Bortolotti et al., \emph{Inverse Problems}, 33(1), 2016]. The proposed approach allows us to obtain accurate relaxation time distributions while keeping short the computation time. Results of numerical experiments on synthetic and real data prove that the proposed method is efficient and effective in reconstructing the peaks and the flat regions that usually characterize NMR relaxation time distributions.

中文翻译：

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使用局部适应多惩罚正则化的 2DNMR 数据反演

二维核磁共振 (NMR) 的一个关键问题是数据反演的速度和准确性。本文提出了一种具有局部适应正则化参数的多重惩罚方法，用于快速准确地反演二维核磁共振数据。该方法解决了一个无约束优化问题，其目标包含一个数据拟合项、一个单$L1$惩罚参数和一个多参数$L2$惩罚。我们建议采用快速迭代收缩和阈值 (FISTA) 方法来解决多重惩罚最小化问题，并提出一个自动程序来计算所有惩罚参数。此过程概括了 [Bortolotti 等人，\emph{Inverse Problems}, 33(1), 2016] 中引入的统一惩罚原则。所提出的方法使我们能够获得准确的弛豫时间分布，同时保持较短的计算时间。合成数据和真实数据的数值实验结果证明，所提出的方法在重建通常表征 NMR 弛豫时间分布的峰和平坦区域方面是有效和有效的。