Robust Variational Approaches to Positivity-Constrained Image Deconvolution

Two approaches to the combination of robust variational deconvolution with positivity constraints are considered. The first approach modifies a standard robust variational deconvolution method by carrying out a gradient descent with respect to a multiplicative perturbation, which can also be considered as gradient descent in a hyperbolic metric. The second approach is based on the representation of the well-known Richardson-Lucy iterative deconvolution as fixed-point iteration for the minimisation of an information divergence functional, again under a multiplicative perturbation model. The asymmetric penaliser function contained in this functional is then varied into a robust penaliser, and complemented with a regulariser. The resulting functional gives rise to a fixed point iteration that we call robust and regularised Richardson-Lucy deconvolution. It achieves an image restoration quality comparable to state-of-the-art variational deconvolution with a computational efficiency similar to that of the original Richardson-Lucy method. Experiments on synthetic and real-world image data demonstrate the performance of the proposed methods.