L₁, L_p, L₂, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

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 L₂ 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 L₁ penalty enforces a greater degree of sparsity than L₂, 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 L₁ and L₂ penalties, may be appropriate for distributions consisting of both narrow and broad components. We evaluate the L₁, L₂, L_p, and EN penalties for model relaxation time distributions consisting of two Gaussian peaks. For distributions with narrow Gaussian peaks, the L₁ penalty works well to maintain sparsity and promote resolution, while the conventional L₂ penalty performs best for distributions with broader peaks. Finally, the L_p and EN penalties do in fact outperform the L₁ and L₂ 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 T₂ distributions from spin echo decay data as well as distributions of other MR parameters, such as apparent diffusion constant, from their multiexponential decay signals.