Local behavior of sparse analysis regularization: Applications to risk estimation

In this paper, we aim at recovering an unknown signal x0 from noisy measurements y=Φx0+w, where Φ is an ill-conditioned or singular linear operator and w accounts for some noise. To regularize such an ill-posed inverse problem, we impose an analysis sparsity prior. More precisely, the recovery is cast as a convex optimization program where the objective is the sum of a quadratic data fidelity term and a regularization term formed of the ℓ1-norm of the correlations between the sought after signal and atoms in a given (generally overcomplete) dictionary. The ℓ1-sparsity analysis prior is weighted by a regularization parameter λ>0. In this paper, we prove that any minimizer of this problem is a piecewise-affine function of the observations y and the regularization parameter λ. As a byproduct, we exploit these properties to get an objectively guided choice of λ. In particular, we develop an extension of the Generalized Stein Unbiased Risk Estimator (GSURE) and show that it is an unbiased and reliable estimator of an appropriately defined risk. The latter encompasses special cases such as the prediction risk, the projection risk and the estimation risk. We apply these risk estimators to the special case of ℓ1-sparsity analysis regularization. We also discuss implementation issues and propose fast algorithms to solve the ℓ1-analysis minimization problem and to compute the associated GSURE. We finally illustrate the applicability of our framework to parameter(s) selection on several imaging problems.