A theoretical framework for the regularization of poisson likelihood estimation problems

Let z = Au + γ be an ill-posed, linear operator equation. Such a model arises, for example, in both astronomical and medical imaging, in which case γ corresponds to background, u the unknown true image, A the forward operator, and z the data. Regularized solutions of this equation can be obtained by solving where T0(Au; z) is the negative-log of the Poisson likelihood functional, and α>0 and J are the regularization parameter and functional, respectively. Our goal in this paper is to determine general conditions which guarantee that Rα defines a regularization scheme for z = Au + γ. Determining the appropriate definition for regularization scheme in this context is important: not only will it serve to unify previous theoretical arguments in this direction, it will provide a framework for future theoretical analyses. To illustrate the latter, we end the paper with an application of the general framework to a case in which an analysis has not been done.