  Radio interferometric (RI) data are noisy under-sampled spatial Fourier
components of the unknown radio sky affected by direction-dependent antenna
gains. Failure to model these antenna gains accurately results in a radio sky
estimate with limited fidelity and resolution. The RI inverse problem has been
recently addressed via a joint calibration and imaging approach which consists
in solving a non-convex minimisation task, involving suitable priors for the
DDEs, namely temporal and spatial smoothness, and sparsity for the unknown
radio map via an $\ell_1$-norm prior, in the context of realistic RI
simulations. Building on these developments, we propose to promote sparsity of
the radio map via a log-sum prior, enforcing sparsity more strongly than the
$\ell_1$-norm. The resulting minimisation task is addressed via a sequence of
non-convex minimisation tasks composed of re-weighted $\ell_1$ image priors,
which are solved approximately. We demonstrate the efficiency of the approach
on RI observations of the celebrated radio galaxy Cygnus~A obtained with the
Karl G. Jansky Very Large Array at X, C, and S bands. More precisely, we
showcase that the approach enhances data fidelity significantly while achieving
high resolution high dynamic range radio maps, confirming the suitability of
the priors considered for the unknown DDEs and radio image. As a clear
qualitative indication of the high fidelity achieved by the data and the
proposed approach, we report the detection of three background sources in the
vicinity of Cyg~A, at S band.
