  In recent works, compressed sensing (CS) and convex optimization techniques
have been applied to radio-interferometric imaging showing the potential to
outperform state-of-the-art imaging algorithms in the field. We review our
latest contributions, which leverage the versatility of convex optimization to
both handle realistic continuous visibilities and offer a highly parallelizable
structure paving the way to significant acceleration of the reconstruction and
high-dimensional data scalability. The new algorithmic structure promoted in a
new software PURIFY (beta version) relies on the simultaneous-direction method
of multipliers (SDMM). The performance of various sparsity priors is evaluated
through simulations in the continuous visibility setting, confirming the
superiority of our recent average sparsity approach SARA.
