Iterative image deconvolution using overcomplete representations

We consider the problem of deconvolving an image with a priori information on its representation in a frame. Our variational approach consists of minimizing the sum of a residual energy and a separable term penalizing each frame coefficient individually. This penalization term may model various properties, in particular sparsity. A general iterative method is proposed and its convergence is established. The novelty of this work is to extend existing methods on two distinct fronts. First, a broad class of convex functions are allowed in the penalization term which, in turn, yields a new class of soft thresholding schemes. Second, while existing results are restricted to orthonormal bases, our algorithmic framework is applicable to much more general overcomplete representations. Numerical simulations are provided.