The University of Montana

Department of Mathematical Sciences

Technical report #25/2008

A Theoretical Framework for the Regularization of
Poisson Likelihood Estimation Problems

**John Bardsley**

University of Montana

**Abstract**

Let *z* = *Au* + γ, where γ > 0 is constant, 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 light intensity. Regularized solutions of this equation can be obtained by solving

where *T*_{0}(Au; z) is the negative-log of the Poisson likelihood functional, and α > 0 and *J* are the regularization parameter and functiosnal, respectively. This variational problem can be motivated from the fact that typical image data contains Poisson noise, and it has been analyzed, for three different choices of *J*, in previous work of the author. Our goal in this paper is to prove that these previous results imply that the approach defines a * regularization scheme*—which we rigorously define here—for each choice of *J*. Determining the appropriate definition for *regularization scheme* in this context is important: not only will it serve to unify the previously mentioned theoretical arguments, it will provide a framework for future theoretical analysis. In addition, we modify our presentation somewhat in order to improve understandability and provide missing arguments from our previous analysis.
**Keywords:** regularization, Poisson likelihood, statistical estimation, mathematical imaging

**AMS Subject Classification:** 65J22, 65K10, 65F22