Iterative weighted risk estimation for nonlinear image restoration with analysis priors

Image acquisition systems invariably introduce blur, which necessitates the use of deblurring algorithms for image restoration. Restoration techniques involving regularization require appropriate selection of the regularization parameter that controls the quality of the restored result. We focus on the problem of automatic adjustment of this parameter for nonlinear image restoration using analysis-type regularizers such as total variation (TV). For this purpose, we use two variants of Stein's unbiased risk estimate (SURE), Predicted-SURE and Projected-SURE, that are applicable for parameter selection in inverse problems involving Gaussian noise. These estimates require the Jacobian matrix of the restoration algorithm evaluated with respect to the data. We derive analytical expressions to recursively update the desired Jacobian matrix for a fast variant of the iterative reweighted least-squares restoration algorithm that can accommodate a variety of regularization criteria. Our method can also be used to compute a nonlinear version of the generalized cross-validation (NGCV) measure for parameter tuning. We demonstrate using simulations that Predicted-SURE, Projected-SURE, and NGCV-based adjustment of the regularization parameter yields near-MSE-optimal results for image restoration using TV, an analysis-type 1-regularization, and a smooth convex edge-preserving regularizer.