Selection of Regularization Parameters for Tikhonov Regularization Applied to Myelin Mapping of the Central Nervous System

Tikhonov regularization of non-linear least squares problems is an unconventional approach that can lead to substantial decreases in the mean squared error of parameter estimates. In previous work we have applied this technique to myelin mapping in the central nervous system with magnetic resonance relaxometry using the bi-exponential model for transverse signal decay in a spin-echo pulse sequence. This is based on the difference in molecular mobility between water trapped in myelin sheaths and non-myelin-associated water. For that study, we used the classical method of generalized cross-validation (GCV) for the selection of the regularization parameter, λ, although other methods, including the discrepancy principle (DP) and the L-curve (LC), are also available. However, both for this problem and more generally, there are no established heuristics for defining the optimal approach for selecting λ. Accordingly, we evaluated the MSE for parameter estimation in biexponential decay, using simulated noisy data across a range of parameter values and signal-to-noise ratios, with λ selection via GCV, the LC and DP, and compared results with an oracle method. Our current work incorporates λ selection using a feed-forward neural network. We will further apply this approach to other signal models arising in magnetic resonance as well as to inverse problems more generally.