%0 Journal Article %1 bardsley2010theoretical %A Bardsley, Johnathan M. %D 2010 %J Inverse Problems and Imaging %K ill-posed_problems inverse_problems %N 1 %P 11-17 %T A theoretical framework for the regularization of Poisson likelihood estimation problems %U http://www.math.umt.edu/bardsley/papers/PoissonGeneralFramework.pdf %V 4 %X 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.

@article{bardsley2010theoretical, 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_{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. }, added-at = {2012-02-25T07:39:13.000+0100}, author = {Bardsley, Johnathan M.}, biburl = {https://www.bibsonomy.org/bibtex/2d94e12329114dff4a9c28270ced2e502/peter.ralph}, interhash = {03eec24b0367451919eadd05cf46cf30}, intrahash = {d94e12329114dff4a9c28270ced2e502}, journal = {Inverse Problems and Imaging}, keywords = {ill-posed_problems inverse_problems}, month = {February}, number = 1, pages = {11-17}, timestamp = {2012-02-25T07:39:13.000+0100}, title = {A theoretical framework for the regularization of Poisson likelihood estimation problems}, url = {http://www.math.umt.edu/bardsley/papers/PoissonGeneralFramework.pdf}, volume = 4, year = 2010 }