This thesis is devoted to the study and the resolution of certains nonlinear problems in signal and image processing via convex analysis. We propose a unified variational investigation of inverse problems and signal decomposition problems which have so far been studied individually, because of their apparent disparity. In the model we adopt, this family of problems is reduced generically to the minimization of the sum of two convex functions with certain regularity properties. Existence, uniqueness and characterization results are obtained for this problem. The proximity operator, introduced by Moreau in 1962 to study certains problems in mechanics, plays a basic role in our analysis. We apply it in particular to obtain new nonlinear signal decomposition schemes. Moreover, this tool is at the heart of the forwardbackward algorithm which we propose to solve the generic problem. This theoretical framework is applied to signal analysis and to image restoration. The restoration problems under consideration are posed on frames and our approach makes it possible to take into account sparsity constraints or to model Bayesian formulations with a priori knowledge on the distribution of the coefficients of the decomposition. Numerical results are provided.
