The University of Montana
Department of Mathematical Sciences

Technical report #23/2008

An Analysis of Regularization by Diffusion for Ill-Posed
Poisson Likelihood Estimation


Johnathan M. Bardsley
The University of Montana

&

N'djekornom Laobeul
The University of Montana

Abstract

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 of the Poisson likelihood function in place of the ubiquitous least squares fit-to-data. However if the underlying mathematical model is assumed to have the form z = Au, where z is the data and A is a linear, compact operator, minimizing the negative-log of the 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 regularization in Bardsley and Laobeul 2008, and total variation regularization in Bardsley and Luttman 2008. In this paper, 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.

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