  Ill-posed linear inverse problems (ILIP), such as restoration and
reconstruction, are a core topic of signal/image processing. A standard
approach to deal with ILIP uses a constrained optimization problem, where a
regularization function is minimized under the constraint that the solution
explains the observations sufficiently well. The regularizer and constraint are
usually convex; however, several particular features of these problems (huge
dimensionality, non-smoothness) preclude the use of off-the-shelf optimization
tools and have stimulated much research. In this paper, we propose a new
efficient algorithm to handle one class of constrained problems (known as basis
pursuit denoising) tailored to image recovery applications. The proposed
algorithm, which belongs to the category of augmented Lagrangian methods, can
be used to deal with a variety of imaging ILIP, including deconvolution and
reconstruction from compressive observations (such as MRI). Experiments testify
for the effectiveness of the proposed method.
