Résumé en anglais

This paper studies the recovery of an unknown signal $x_0$ from low dimensional noisy observations $y = \Phi x_0 + w$, where $\Phi$ is an ill-posed linear operator and $w$ accounts for some noise. We focus our attention to sparse analysis regularization. The recovery is performed by minimizing the sum of a quadratic data fidelity term and the $\lun$-norm of the correlations between the sought after signal and atoms in a given (generally overcomplete) dictionary. The $\lun$ prior is weighted by a regularization parameter $\lambda > 0$ that accounts for the noise level. In this paper, we prove that minimizers of this problem are piecewise-affine functions of the observations $y$ and the regularization parameter $\lambda$. As a byproduct, we exploit these properties to get an objectively guided choice of $\lambda$. More precisely, we propose an extension of the Generalized Stein Unbiased Risk Estimator (GSURE) and show that it is an unbiased estimator of an appropriately defined risk. This encompasses special cases such as the prediction risk, the projection risk and the estimation risk. We also discuss implementation issues and propose fast algorithms. We apply these risk estimators to the special case of sparse analysis regularization. We finally illustrate the applicability of our framework on several imaging problems.