An analysis of regularization by diffusion for ill-posed Poisson likelihood estimations

The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative-log Poisson likelihood function in place of the least-squares fit-to-data function. However, if the underlying mathematical model is assumed to have the form z= Au+γ, where z is the data and A is a compact operator and γ is the background light intensity, minimizing the negative-log Poisson likelihood function is an ill-posed problem, and hence some form of regularization is required. In previous work, the authors have performed theoretical analyses of two approaches for regularization in this setting: standard Tikhonov and total variation regularization. In this article, we consider a class of regularization functionals defined by differential operators of diffusion type, and our main results constitute a theoretical justification of this approach. However, in order to demonstrate that the approach is effective in practice, we follow our theoretical analysis with a numerical experiment.