The University of Montana

Department of Mathematical Sciences

Technical report #16/2006

Tikhonov Regularization for Ill-Posed Poisson Likelihood Estimation: Analysis and Computation

**Johnathan M. Bardsley and N'djekornom Laobeul**

Department of Mathematical Sciences

The University of Montana (USA)

**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 logarithm of the Poisson likelihood 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 *A* is
a linear, compact operator, Poisson likelihood estimation is
ill-posed, and hence some form of regularization is required. In
a recent paper by the first author, a numerical method is presented and analyzed for
Tikhonov regularized Poisson likelihood estimation, but no
theoretical justification of the approach is given. Our primary
objective in this paper is to provide such a theoretical
justification. We then briefly present a computational method of
that is very effective and computationally
efficient for this problem. The practical validity of the approach
is then demonstrated on a synthetic example from astronomical
imaging.

**Keywords:** regularization, ill-posed
problems, maximum likelihood estimation, image reconstruction,
nonnegatively constrained minimization

**AMS Subject Classification:** 65J22, 65K10, 65F22

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