Abstract
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_u0 \T_0(Au;z)+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.
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