Off-the-grid regularisation for Poisson inverse problems

Off-the-grid regularisation has been extensively employed over the last decadein the context of ill-posed inverse problems formulated in the continuous settingof the space of Radon measures M(Ω). These approaches enjoy convexity andcounteract the discretisation biases as well the numerical instabilities typical oftheir discrete counterparts. In the framework of sparse reconstruction of discretepoint measures (sum of weighted Diracs), a Total Variation regularisation normin M(Ω) is typically combined with an L2 data term modelling additive Gaussiannoise. To assess the framework of off-the-grid regularisation in the presenceof signal-dependent Poisson noise, we consider in this work a variational modelwhere Total Variation regularisation is coupled with a Kullback-Leibler data termunder a non-negativity constraint. Analytically, we study the optimality conditionsof the composite functional and analyse its dual problem. Then, we consideran homotopy strategy to select an optimal regularisation parameter and use itwithin a Sliding Frank-Wolfe algorithm. Several numerical experiments on both1D/2D/3D simulated and real 3D fluorescent microscopy data are reported.