A THEORETICAL FRAMEWORK FOR THE REGULARIZATION OF POISSON LIKELIHOOD ESTIMATION PROBLEMS

Let z = Au + gamma be an ill-posed, linear operator equation. Such a model arises, for example, in both astronomical and medical imaging, in which case gamma 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 R(alpha)(A,z) = arg min(u >= 0){T(0)(Au;z) + alpha J(u)}, where T(0) (Au;z) is the negative-log of the Poisson likelihood functional, and alpha > 0 and J are the regularization parameter and functional, respectively. Our goal in this paper is to determine general conditions which guarantee that R(alpha) defines a regularization scheme for z = Au + gamma. 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 frame work to a case in which an analysis has not been done.