Variational methods are widely used in image processing. They allow us to propose models taking into account specificities of the tackled problems. They also enable the study of the properties of solutions. The proposed functional in image processing are not smooth (to account for the presence of interfaces) and not necessarily convex. This naturally raises questions of existence of solutions, uniqueness, and (fast) computation. The choice of the regularization is often based on an assumption of sparsity (eg, of the gradient, or in a transformed domain). Non-local interactions and the concept of patch are typically involved in the design of functional. Finally, all these approaches are based in principle on the weighting of the data fidelity term and the regularization term, which leads to the question of the reliable estimation of this parameter.
Charles Dossal proved in 2015 with Antonin Chambolle the convergence of an extremely popular algorithm since its introduction in 2008 by Beck and Teboulle (for being extremely efficient in practice), the FISTA algorithm (Fast Iterative Shrinkage Algorithm). The convergence of the algorithm has remained an open problem in the communities of optimization/compressed sensing/image processing/machine learning for 7 years until Charles Dossal work.