  In traditional optical imaging systems, the spatial resolution is limited by
the physics of diffraction, which acts as a low-pass filter. The information on
sub-wavelength features is carried by evanescent waves, never reaching the
camera, thereby posing a hard limit on resolution: the so-called diffraction
limit. Modern microscopic methods enable super-resolution, by employing
florescence techniques. State-of-the-art localization based fluorescence
subwavelength imaging techniques such as PALM and STORM achieve sub-diffraction
spatial resolution of several tens of nano-meters. However, they require tens
of thousands of exposures, which limits their temporal resolution. We have
recently proposed SPARCOM (sparsity based super-resolution correlation
microscopy), which exploits the sparse nature of the fluorophores distribution,
alongside a statistical prior of uncorrelated emissions, and showed that
SPARCOM achieves spatial resolution comparable to PALM/STORM, while capturing
the data hundreds of times faster. Here, we provide a detailed mathematical
formulation of SPARCOM, which in turn leads to an efficient numerical
implementation, suitable for large-scale problems. We further extend our method
to a general framework for sparsity based super-resolution imaging, in which
sparsity can be assumed in other domains such as wavelet or discrete-cosine,
leading to improved reconstructions in a variety of physical settings.
