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