February  2010, 4(1): 11-17. doi: 10.3934/ipi.2010.4.11

A theoretical framework for the regularization of Poisson likelihood estimation problems

1. 

Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812, United States

Received  October 2008 Revised  October 2009 Published  February 2010

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

$R_\alpha(A,z)= arg\min_{u\geq 0} \{T_0(Au;z)+\alpha J(u)\},$

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 framework to a case in which an analysis has not been done.

Citation: Johnathan M. Bardsley. A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Problems & Imaging, 2010, 4 (1) : 11-17. doi: 10.3934/ipi.2010.4.11
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