Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

Abstract

This paper deals with the restoration of images corrupted by a non-invertible or ill-conditioned linear transform and Poisson noise. Poisson data typically occur in imaging processes where the images are obtained by counting particles, e.g., photons, that hit the image support. By using the Anscombe transform, the Poisson noise can be approximated by an additive Gaussian noise with zero mean and unit variance. Then, the least squares difference between the Anscombe transformed corrupted image and the original image can be estimated by the number of observations. We use this information by considering an Anscombe transformed constrained model to restore the image. The advantage with respect to corresponding penalized approaches lies in the existence of a simple model for parameter estimation. We solve the constrained minimization problem by applying a primal-dual algorithm together with a projection onto the epigraph of a convex function related to the Anscombe transform. We show that this epigraphical projection can be efficiently computed by Newton’s methods with an appropriate initialization. Numerical examples demonstrate the good performance of our approach, in particular, its close behaviour with respect to the I-divergence constrained model.