  In a recent article series, the authors have promoted convex optimization
algorithms for radio-interferometric imaging in the framework of compressed
sensing, which leverages sparsity regularization priors for the associated
inverse problem and defines a minimization problem for image reconstruction.
This approach was shown, in theory and through simulations in a simple discrete
visibility setting, to have the potential to outperform significantly CLEAN and
its evolutions. In this work, we leverage the versatility of convex
optimization in solving minimization problems 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 relies on the
simultaneous-direction method of multipliers (SDMM), and contrasts with the
current major-minor cycle structure of CLEAN and its evolutions, which in
particular cannot handle the state-of-the-art minimization problems under
consideration where neither the regularization term nor the data term are
differentiable functions. We release a beta version of an SDMM-based imaging
software written in C and dubbed PURIFY (http://basp-group.github.io/purify/)
that handles various sparsity priors, including our recent average sparsity
approach SARA. We evaluate the performance of different priors through
simulations in the continuous visibility setting, confirming the superiority of
SARA.
