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

Abstract

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