Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization

Although much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications, applying the state of the art regularizers (such as those based on wavelets or total variation) to this class of images is still an open research front. This paper proposes a new approach to deconvolving Poissonian images, with the following building blocks: (a) the standard regularization/ maximum a posteriori (MAP) criterion, combining the Poisson log-likelihood with a regularizer (log-prior) is adopted; (b) the resulting optimization problem (which is hard, because the log-likelihood is non-quadratic and nonseparable and the regularizer is non-smooth) is transformed into an equivalent constrained problem, by a variable splitting procedure; (c) an augmented Lagrangian method is used to address this constrained problem. The resulting algorithm is shown to outperform alternative state-of-the-art methods.