Off-the-Grid Curve Reconstruction through Divergence Regularization: An Extreme Point Result.

We propose a new strategy for curve reconstruction in an image through an off-the-grid variational framework, inspired by spike reconstruction in the literature. We introduce a new functional CROC on the space of 2-dimensional Radon measures with finite divergence denoted ', and we estab-lish several theoretical tools through the definition of a certificate. Our main contribution lies in the sharp characterization of the extreme points of the unit ball of the '-norm: there are exact measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise charac-terization of our functional minimizers and further opening a promising avenue for the algorithmic implementation.