Regularization Parameter Selection for Nonlinear Iterative Image Restoration and MRI Reconstruction Using GCV and SURE-Based Methods

Regularized iterative reconstruction algorithms for imaging inverse problems require selection of appropriate regularization parameter values. We focus on the challenging problem of tuning regularization parameters for nonlinear algorithms for the case of additive (possibly complex) Gaussian noise. Generalized cross-validation (GCV) and (weighted) mean-squared error (MSE) approaches [based on Stein´s unbiased risk estimate (SURE)] need the Jacobian matrix of the nonlinear reconstruction operator (representative of the iterative algorithm) with respect to the data. We derive the desired Jacobian matrix for two types of nonlinear iterative algorithms: a fast variant of the standard iterative reweighted least-squares method and the contemporary split-Bregman algorithm, both of which can accommodate a wide variety of analysis- and synthesis-type regularizers. The proposed approach iteratively computes two weighted SURE-type measures: predicted-SURE and projected-SURE (which require knowledge of noise variance σ²), and GCV (which does not need σ²) for these algorithms. We apply the methods to image restoration and to magnetic resonance image (MRI) reconstruction using total variation and an analysis-type ℓ₁-regularization. We demonstrate through simulations and experiments with real data that minimizing predicted-SURE and projected-SURE consistently lead to near-MSE-optimal reconstructions. We also observe that minimizing GCV yields reconstruction results that are near-MSE-optimal for image restoration and slightly suboptimal for MRI. Theoretical derivations in this paper related to Jacobian matrix evaluations can be extended, in principle, to other types of regularizers and reconstruction algorithms.