Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation
Idioma
en
Article de revue
Este ítem está publicado en
Applied and Computational Harmonic Analysis. 2013, vol. 35, n° 3, p. 433-451
Elsevier
Resumen en inglés
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 ...Leer más >
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.< Leer menos
Palabras clave en inglés
sparsity
analysis regularization
inverse problems
L1 minimization
local variation
degrees of freedom
SURE
GSURE
unbiased risk estimation
Proyecto europeo
Sparsity, Image and Geometry to Model Adaptively Visual Processings
Proyecto ANR
Adaptivité pour la représentation des images naturelles et des textures - ANR-08-EMER-0009
Orígen
Importado de HalCentros de investigación