A continuous, non-convex & sparse super-resolution approach for fluorescence microscopy data with Poisson noise

We propose a non-convex sparsity-promoting variational model for the problem of super-resolution in Single Molecule Localization Microscopy (SMLM). Namely, we study a continuous non-convex relaxation of a non-continuous and non-convex variational model where a weighted-L2 data fidelity modeling signal-dependent Poisson noise is combined with an L0-regularization to promote signal sparsity. The proposed relaxation is obtained by adapting the Continuous Exact L0 (CEL0) relaxation of the analogous `2`0 problem with Gaussian noise to the Poisson scenario, which is more realistic in fluorescence microscopy applications. The associated optimization problem is then solved by an iterative reweighted L1 (IRL1) algorithm. The weighted-L2 data fidelity leads to a challenging estimation of the algorithmic parameters for which efficient computation strategies are detailed. To validate our approach, we report qualitative and quantitative localization results for a simulated dataset, showing that the proposed weighted-CEL0 (WCEL0) model is well suited and capable to deal with Poisson measurements with high accuracy and precision.