Local behavior of sparse analysis regularization: Applications to risk estimation

In this paper, we aim at recovering an unknown signal x(0) from noisy measurements y = Phi x(0) + w, where Phi 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 l(1)-norm of the correlations between the sought after signal and atoms in a given (generally overcomplete) dictionary. The l(1)-sparsity analysis prior is weighted by a regularization parameter lambda > 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 lambda. As a byproduct, we exploit these properties to get an objectively guided choice of lambda. 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 l(1)-sparsity analysis regularization. We also discuss implementation issues and propose fast algorithms to solve the l(1)-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. (C) 2012 Elsevier Inc. All rights reserved.