Unbiased risk estimation for sparse analysis regularization

In this paper, we propose a rigorous derivation of the expression of the projected Generalized Stein Unbiased Risk Estimator (GSURE) for the estimation of the (projected) risk associated to regularized ill-posed linear inverse problems using sparsity-promoting ℓ¹ penalty. The projected GSURE is an unbiased estimator of the recovery risk on the vector projected on the orthogonal of the degradation operator kernel. Our framework can handle many well-known regularizations including sparse synthesis- (e.g. wavelet) and analysis-type priors (e.g. total variation). A distinctive novelty of this work is that, unlike previously proposed ℓ¹ risk estimators, we have a closed-form expression that can be implemented efficiently once the solution of the inverse problem is computed. To support our claims, numerical examples on ill-posed inverse problems with analysis and synthesis regularizations are reported where our GSURE estimates are used to tune the regularization parameter.