Representing the image to be inpainted in an appropriate sparse dictionary, and combining elements from bayesian statistics, we introduce an expectation-maximization (EM) algorithm for image inpainting. From a statistical point of view, the inpainting can be viewed as an estimation problem with missing data. Towards this goal, we propose the idea of using the EM mechanism in a bayesian framework, where a sparsity promoting prior penalty is imposed on the reconstructed coefficients. The EM framework gives a principled way to establish formally the idea that missing samples can be recovered based on sparse representations. We first introduce an easy and efficient sparse-representation-based iterative algorithm for image inpainting. A side effect of such an algorithm is decomposition of the inpainted image in different mophological components as recently introduced by Starck et al. We give a formal description of the problem and derive its theoretical convergence properties for a wide class of penalties. Particularly, we establish that it converges in a strong sense, and give sufficient conditions for convergence to a local or a global minimum. Compared to its competitors, this algorithms allows a high degree of flexibility to recover different structural components in the image (piece-wise smooth, curvilinear, texture, etc). We also describe some ideas to automatically find the regularization parameter.