  Compressed sensing theory is slowly making its way to solve more and more
astronomical inverse problems. We address here the application of sparse
representations, convex optimization and proximal theory to radio
interferometric imaging. First, we expose the theory behind interferometric
imaging, sparse representations and convex optimization, and second, we
illustrate their application with numerical tests with SASIR, an implementation
of the FISTA, a Forward-Backward splitting algorithm hosted in a LOFAR imager.
Various tests have been conducted in Garsden et al., 2015. The main results
are: i) an improved angular resolution (super resolution of a factor ~2) with
point sources as compared to CLEAN on the same data, ii) correct photometry
measurements on a field of point sources at high dynamic range and iii) the
imaging of extended sources with improved fidelity. SASIR provides better
reconstructions (five time less residuals) of the extended emission as compared
to CLEAN. With the advent of large radiotelescopes, there is scope for
improving classical imaging methods with convex optimization methods combined
with sparse representations.
