An augmented Lagrangian approach to linear inverse problems with compound regularization

In some imaging inverse problems, it may be desired that the solution simultaneously exhibits a set of properties not enforceable by a single regularizer. To attain this goal, one may use a linear combinations of regularizers, thus encouraging the solution to simultaneously exhibit the characteristics enforced by each of them. This paper addresses the optimization problem associated with this type of compound regularization, using an alternating direction optimization algorithm. We illustrate the approach in two image deblurring problems - one in which the images are simultaneously sparse and piece-wise smooth, using a linear combination of the ℓ₁ and total variation regularizers, and the other for a natural image with a combination of frame-based synthesis and analysis ℓ₁ norm regularizers.