Proof of the resolution-doubling of Random Illumination Microscopy using the variance of the speckled images

Owing to minor modifications of the optical setup, Random Illumination Microscopy (RIM) surpasses the resolution limit of a standard fluorescence microscope. RIM uses speckle illuminations of the sample to derive a single variance image from the resulting diffraction-limited acquisitions. Variance-matching iterations then produce a super-resolved estimate of the sample. Here, we demonstrate that in the noiseless case, variance-matching yields a unique solution for the set of spatial frequencies corresponding to a doubled resolution limit. A similar result was already proven for covariance-matching, but covariance-based iterations are not implementable in practice, due to the huge size of the covariance matrix and to the induced numerical complexity. Our new identifiability result is a strong theoretical evidence supporting the super-resolution capability of the variance-matching version of RIM.