Tikhonov regularized Poisson likelihood estimation: theoretical justification and a computational method

The noise contained in images collected by a charge coupled device 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 Bardsley, J.M. and Vogel, C.R., 2004, A nonnegatively constrained convex programming method for image reconstruction. SIAM Journal on Scientific Computing, 25(4), pp. 1326–1343, 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 article is to provide such a theoretical justification. We then briefly present the computational method of Bardsley, J.M. and Vogel, C.R., 2004, A nonnegatively constrained convex programming method for image reconstruction. SIAM Journal on Scientific Computing, 25(4), pp. 1326–1343, which 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.