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International audienceIn this paper, we propose two algorithms to solve a large class of linear inverse problems when the observations are corrupted by various types of noises. A proper data fidelity term (log-likelihood) is introduced to reflect the statistics of the noise (e.g. Gaussian, Poisson, multiplicative, etc.) independently of the degradation operator. On the other hand, the regularization is constructed through different terms reflecting a priori knowledge on the images. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a composite non-smooth convex objective functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers for different kind of noises. Then we solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution, inpainting and denoising with some comparison to prior methods are also reported

Linear inverse problems with various noise models and mixed regularizations