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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.

Gridless sparse spike reconstruction is a rather new research field with significant results for the super-resolution problem, where we want to retrieve fine-scale details from a noisy and filtered acquisition. To tackle this problem, we are interested in optimisation under some prior, typically the sparsity i.e., the source is composed of spikes. Following the seminal work on the generalised LASSO for measures called the Beurling-Lasso (BLASSO), we will give a review on the chief theoretical and numerical breakthrough of the off-the-grid inverse problem, as we illustrate its usefulness to the super-resolution problem in Single Molecule Localisation Microscopy (SMLM) through new reconstruction metrics and tests on synthetic and real SMLM data we performed for this review.

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