Restoration of edges by minimizing non-convex cost-functions

We consider the restoration of discrete signals and images using least-squares with non-convex regularization. Our goal is to understand how the restoration of edges at the (local) minimizers of the cost function is determined by the shape of the regularization term. This question is of paramount importance for a relevant choice of regularization term. We show that the differences between neighboring pixels in homogeneous regions are smaller than a small threshold while they are larger than a large threshold at edges: we can say that the former are shrunk while the latter are enhanced. This entails a natural classification of differences as smooth or edges. Furthermore, if the original signal or image is a scaled characteristic function of a subset, we show that the global minimizer is smooth everywhere if the contrast is low, whereas edges are correctly recovered at higher (finite) contrast. Explicit expressions are derived for the truncated quadratic and the "0-1" regularization function. It is seen that restoration using non-convex regularization is fundamentally different from edge-preserving convex regularization. Our theoretical results are illustrated using a numerical experiment.