Weighted-CEL0 sparse regularisation for molecule localisation in super-resolution microscopy with Poisson data

We consider a variational model for Single Molecule Localisation Microscopy (SMLM) super-resolution. More specifically, we study a generalization of the Continuous Exact $\ell_{0}$ (CELO) penalty, recently introduced to relax the $\ell_{2}-\ell_{0}$ problem, where a weighted-$\ell_{2}$ data fidelity now models signal-dependent Poisson noise. For the numerical solution of the associated non-convex minimisation problem, we propose an iterative reweighted $\ell_{1}$ (IRL1) algorithm, for which efficient parameter computation strategies are detailed. Both qualitative and quantitative molecule localisation results are reported, showing that the proposed weighted-CELO (wCELO) model for Poisson noisy data improves the results obtained by CELO and state-of-the art deep-learning approaches for the high-density SMLM ISBI 2013 dataset.