Already since the work by Abbe and Rayleigh the difficulty of super resolution where one wants to recover a collection of point sources from low-resolved microscopy measurements is thought to be dependent on whether the distance between the sources is below or above a certain resolution or diffraction limit. Even though there has been a number of approaches to define this limit more rigorously, there is still a gap between the situation where the task is known to be hard and scenarios where the task is provably simpler. For instance, an interesting approach for the univariate case using the size of the Cramér-Rao lower bound was introduced in a recent work by Ferreira Da Costa and Mitra. In this paper, we prove their conjecture on the transition point between good and worse tractability of super resolution and extend it to higher dimensions. Specifically, the bivariate statistical analysis allows to link the findings by the Cramér-Rao lower bound to the classical Rayleigh limit.

Abstract. This paper considers superresolution (SR) as the mapping of Fourier coefficients of a discrete measure on [Formula: see text] to its support and weights. We focus on weak SR assuming a condition on the separation of the involved measures similar to the Rayleigh criterion and prove that the reconstruction map satisfies a local Lipschitz property giving explicit estimates for the Lipschitz constant depending on the dimension [Formula: see text] and the sampling effort while improving a bound on the assumed separation from [ 6 ]. With the Wasserstein distance as the metric on the space of measures we even conclude that weak SR is globally Lipschitz continuous and hence well-conditioned.