Weakly Constrained Minimization: Application to the Estimation of Images and Signals Involving Constant Regions

We focus on the question of how the shape of a cost-function determines the features manifested by its local (and hence global) minimizers. Our goal is to check the possibility that the local minimizers of an unconstrained cost-function satisfy different subsets of affine constraints dependent on the data, hence the word weak. A typical example is the estimation of images and signals which are constant on some regions. We provide general conditions on cost-functions which ensure that their minimizers can satisfy weak constraints when noisy data range over an open subset. These cost-functions are non-smooth at all points satisfying the weak constraints. In contrast, the local minimizers of smooth cost-functions can almost never satisfy weak constraints. These results, obtained in a general setting, are applied to analyze the minimizers of cost-functions, composed of a data-fidelity term and a regularization term. We thus consider the effect produced by non-smooth regularization, in comparison with smooth regularization. In particular, these results explain the stair-casing effect, well known in total-variation methods. Theoretical results are illustrated using analytical examples and numerical experiments.