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By introducing a new operator theory, we provide a unified mathematical theory for general source resolution in the multi-illumination imaging problem. Our main idea is to transform multi-illumination imaging into single-snapshot imaging with a new imaging kernel that depends on both the illumination patterns and the point spread function of the imaging system. We therefore prove that the resolution of multi-illumination imaging is approximately determined by the essential cutoff frequency of the new imaging kernel, which is roughly limited by the sum of the cutoff frequency of the point spread function and the maximum essential frequency in the illumination patterns. Our theory provides a unified way to estimate the resolution of various existing super-resolution modalities and results in the same estimates as those obtained in experiments. In addition, based on the reformulation of the multi-illumination imaging problem, we also estimate the resolution limits for resolving both complex and positive sources by sparsity-based approaches. We show that the resolution of multi-illumination imaging is approximately determined by the new imaging kernel from our operator theory and better resolution can be realized by sparsity-promoting techniques in practice but only for resolving very sparse sources. This explains experimentally observed phenomena in some sparsity-based super-resolution modalities.

International audience; 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. Variancematching iterations then produce a super-resolved estimate of the sample. Here, we demonstrate that in the noiseless case, variancematching 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 covariancebased 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.