Infimal Convolution of Data Discrepancies for Mixed Noise Removal

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 frequently considered in applications, such as salt & pepper and Gaussian, and Gaussian and Poisson noise mixtures. We derive a variational image denoising model that features a total variation regularization term and a data discrepancy encoding 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. 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 methods dealing with the mixed noise case are shown.