  We consider the problem of image denoising in the presence of noise whose
statistical properties are a combination of two different distributions. We
focus on noise distributions that are frequently considered in applications, in
particular mixtures of salt & pepper and Gaussian noise, and Gaussian and
Poisson noise. We derive a variational image denoising model that features a
total variation regularisation term and a data discrepancy that features the
mixed noise as an infimal convolution of discrepancy terms of the single-noise
distributions. We give a statistical derivation of this model by joint Maximum
A-Posteriori (MAP) estimation, and discuss in particular its interpretation as
the MAP of a so-called infinity convolution of two noise distributions.
Moreover, classical single-noise models are recovered asymptotically as the
weighting parameters go to infinity. The numerical solution of the model is
computed using second order Newton-type methods. Numerical results show the
decomposition of the noise into its constituting components. The paper is
furnished with several numerical experiments and comparisons with other
existing methods dealing with the mixed noise case are shown.
