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 5 framework, inspired by spike reconstruction in the literature. We introduce a new functional CROC 6 on the space of 2-dimensional Radon measures with finite divergence denoted (cid:86) , and we estab-7 lish several theoretical tools through the definition of a certificate. Our main contribution lies in 8 the sharp characterisation of the extreme points of the unit ball of the (cid:86) -norm: there are exactly 9 measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise charac-10 terisation of our functional minimisers and further opening a promising avenue for the algorithmic 11 implementation.