  Radio interferometric imaging aims to estimate an unknown sky intensity image
from degraded observations, acquired through an antenna array. In the
theoretical case of a perfectly calibrated array, it has been shown that
solving the corresponding imaging problem by iterative algorithms based on
convex optimization and compressive sensing theory can be competitive with
classical algorithms such as CLEAN. However, in practice, antenna-based gains
are unknown and have to be calibrated. Future radio telescopes, such as the
SKA, aim at improving imaging resolution and sensitivity by orders of
magnitude. At this precision level, the direction-dependency of the gains must
be accounted for, and radio interferometric imaging can be understood as a
blind deconvolution problem. In this context, the underlying minimization
problem is non-convex, and adapted techniques have to be designed. In this
work, leveraging recent developments in non-convex optimization, we propose the
first joint calibration and imaging method in radio interferometry, with proven
convergence guarantees. Our approach, based on a block-coordinate
forward-backward algorithm, jointly accounts for visibilities and suitable
priors on both the image and the direction-dependent effects (DDEs). As
demonstrated in recent works, sparsity remains the prior of choice for the
image, while DDEs are modelled as smooth functions of the sky, i.e. spatially
band-limited. Finally, we show through simulations the efficiency of our
method, for the reconstruction of both images of point sources and complex
extended sources. MATLAB code is available on GitHub.
