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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Technical Report:** pdf (228 KB)