Either Fit to Data Entries or Locally to Prior: The Minimizers of Objectives with Nonsmooth Nonconvex Data Fidelity and Regularization

We investigate coercive objective functions composed of a data-fidelity term and a regularization term. Both of these terms are non differentiable and non convex, at least one of them being strictly non convex. The regularization term is defined on a class of linear operators including finite differences. Their minimizers exhibit amazing properties. Each minimizer is the exact solution of an (overdetermined) linear system composed partly of linear operators from the data term, partly of linear operators involved in the regularization term. This is a strong property that is useful when we know that some of the data entries are faithful and the linear operators in the regularization term provide a correct modeling of the sought-after image or signal. It can be used to tune numerical schemes as well. Beacon applications include super resolution, restoration using frame representations, inpainting, morphologic component analysis, and so on. Various examples illustrate the theory and show the interest of this new class of objectives.