  (arXiv abridged abstract) The current years are seeing huge developments of
radio telescopes and a tremendous increase of their capabilities. Such systems
make mandatory the design of more sophisticated techniques not only for
transporting, storing and processing this new generation of radio
interferometric data, but also for restoring the astrophysical information
contained in such data. In this paper we present a new radio deconvolution
algorithm named MORESANE and its application to fully realistic simulated data
of MeerKAT, one of the SKA precursors. This method has been designed for the
difficult case of restoring diffuse astronomical sources which are faint in
brightness, complex in morphology and possibly buried in the dirty beam's side
lobes of bright radio sources in the field. MORESANE is a greedy algorithm
which combines complementary types of sparse recovery methods in order to
reconstruct the most appropriate sky model from observed radio visibilities. A
synthesis approach is used for the reconstruction of images, in which the
synthesis atoms representing the unknown sources are learned using analysis
priors. We apply this new deconvolution method to fully realistic simulations
of radio observations of a galaxy cluster and of an HII region in M31. We show
that MORESANE is able to efficiently reconstruct images composed from a wide
variety of sources from radio interferometric data. Comparisons with other
available algorithms, which include multi-scale CLEAN and the recently proposed
methods by Li et al. (2011) and Carrillo et al. (2012), indicate that MORESANE
provides competitive results in terms of both total flux/surface brightness
conservation and fidelity of the reconstructed model. MORESANE seems
particularly well suited for the recovery of diffuse and extended sources, as
well as bright and compact radio sources known to be hosted in galaxy clusters.
