Two constrained formulations for deblurring Poisson noisy images

Deblurring noisy Poisson images has recently been subject of an increasingly amount of works in many areas such as astronomy or biological imaging. Several methods have promoted explicit prior on the solution to regularize the ill-posed inverse problem and to improve the quality of the image. In each of these methods, a regularizing parameter is introduced to control the weight of the prior. Unfortunately, this regularizing parameter has to be manually set such that it gives the best qualitative results. To tackle this issue, we present in this paper two constrained formulations for the Poisson deconvolution problem, derived from recent advances in regularizing parameter estimation for Poisson noise. We first show how to improve the accuracy of these estimators and how to link these estimators to constrained formulations. We then propose an algorithm to solve the resulting optimization problems and detail how to per form the projections on the constraints. Results on real and synthetic data are presented.