  This paper focuses on efficient computational approaches to compute
approximate solutions of a linear inverse problem that is contaminated with
mixed Poisson--Gaussian noise, and when there are additional outliers in the
measured data. The Poisson--Gaussian noise leads to a weighted minimization
problem, with solution-dependent weights. To address outliers, the standard
least squares fit-to-data metric is replaced by the Talwar robust regression
function. Convexity, regularization parameter selection schemes, and
incorporation of non-negative constraints are investigated. A projected Newton
algorithm is used to solve the resulting constrained optimization problem, and
a preconditioner is proposed to accelerate conjugate gradient Hessian solves.
Numerical experiments on problems from image deblurring illustrate the
effectiveness of the methods.
