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.
Keywords:
AMS Subject Classification:
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