Multi-Regularization Reconstruction of One-Dimensional $T_2$ Distributions in Magnetic Resonance Relaxometry with a Gaussian Basis

We consider the inverse problem of recovering the probability distribution function of $T_2$ relaxation times from NMR transverse relaxometry experiments. This problem is a variant of the inverse Laplace transform and hence ill-posed. We cast this within the framework of a Gaussian mixture model to obtain a least-square problem with an $L_2$ regularization term. We propose a new method for incorporating regularization into the solution; rather than seeking to replace the native problem with a suitable mathematically close, regularized, version, we instead augment the native formulation with regularization. We term this new approach 'multi-regularization'; it avoids the treacherous process of selecting a single best regularization parameter $\lambda$ and instead permits incorporation of several degrees of regularization into the solution. We illustrate the method with extensive simulation results as well as application to real experimental data.