Sparse Representations and EM Algorithm for Joint Image Decomposition and Inpainting

International audienceRepresenting 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