  The spatial resolution of images of living samples obtained by fluorescence
microscopes is physically limited due to the diffraction of visible light,
which makes the study of entities of size less than the diffraction barrier
(around 200 nm in the x-y plane) very challenging. To overcome this limitation,
several deconvolution and super-resolution techniques have been proposed.
Within the framework of inverse problems, modern approaches in fluorescence
microscopy reconstruct a super-resolved image from a temporal stack of frames
by carefully designing suitable hand-crafted sparsity-promoting regularisers.
Numerically, such approaches are solved by proximal gradient-based iterative
schemes. Aiming at obtaining a reconstruction more adapted to sample geometries
(e.g. thin filaments), we adopt a plug-and-play denoising approach with
convergence guarantees and replace the proximity operator associated with the
explicit image regulariser with an image denoiser (i.e. a pre-trained network)
which, upon appropriate training, mimics the action of an implicit prior. To
account for the independence of the fluctuations between molecules, the model
relies on second-order statistics. The denoiser is then trained on covariance
images coming from data representing sequences of fluctuating fluorescent
molecules with filament structure. The method is evaluated on both simulated
and real fluorescence microscopy images, showing its ability to correctly
reconstruct filament structures with high values of peak signal-to-noise ratio
(PSNR).
