Determination of the distribution of magnetic resonance (MR) transverse relaxation times is emerging as an important method for materials characterization, including assessment of tissue pathology in biomedicine. These distributions are obtained from the inverse Laplace transform (ILT) of multiexponential decay data. Stabilization of this classically ill-posed problem is most commonly attempted using Tikhonov regularization with an L 2 penalty term. However, with the availability of convex optimization algorithms and recognition of the importance of sparsity in model reconstruction, there has been increasing interest in alternative penalties. The L 1 penalty enforces a greater degree of sparsity than L 2 , and so may be suitable for highly localized relaxation time distributions. In addition, L p penalties, 1 < p < 2, and the elastic net (EN) penalty, defined as a linear combination of L 1 and L 2 penalties, may be appropriate for distributions consisting of both narrow and broad components. We evaluate the L 1 , L 2 , L p , and EN penalties for model relaxation time distributions consisting of two Gaussian peaks. For distributions with narrow Gaussian peaks, the L 1 penalty works well to maintain sparsity and promote resolution, while the conventional L 2 penalty performs best for distributions with broader peaks. Finally, the L p and EN penalties do in fact outperform the L 1 and L 2 penalties for distributions with components of unequal widths. These findings serve as indicators of appropriate regularization in the typical situation in which the experimentalist has a priori knowledge of the general characteristics of the underlying relaxation time distribution. Our findings can be applied to both the recovery of T 2 distributions from spin echo decay data as well as distributions of other MR parameters, such as apparent diffusion constant, from their multiexponential decay signals. K E Y W O R D Sinverse Laplace transform, inverse problems, NMR relaxometry, non-negative least squares, regularization

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