Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation

In this paper, we aim at recovering an unknown signal x0 from noisy L1measurements y=Phi*x0+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 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 lambda>0. In this paper, we prove that any minimizers 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 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

Core-Shell Columns in High-Performance Liquid Chromatography: Food Analysis Applications

Theincreased separation efficiency provided by the newtechnology of column packed with core-shell particles in high-performance liquid chromatography (HPLC) has resulted in their widespread diffusion in several analytical fields: from pharmaceutical, biological, environmental, and toxicological.The present paper presents their most recent applications in food analysis.Their use has proved to be particularly advantageous for the determination of compounds at trace levels or when a large amount of samples must be analyzed fast using reliable and solvent-saving apparatus. The literature hereby described shows how the outstanding performances provided by core-shell particles column on a traditional HPLC instruments are comparable to those obtained with a costly UHPLC instrumentation, making this novel column a promising key tool in food analysis