A gradient-like variational Bayesian approach for joint image super-resolution and source separation, application to astrophysical map-making

In this work, a new unsupervised Bayesian method for joint image super-resolution and component separation is introduced. More precisely, we are interested in super-resolution for astrophysical map-making and separation between extended and point emissions. This is tackled as an inverse problem in a Bayesian framework, where a Markovian model is used as a prior for the extended emission and a student´s t-distribution is attributed for the point sources component. All model and noise parameters are unknown, therefore we chose to estimate them jointly with the images. Nevertheless, both Joint Maximum A Posteriori (JMAP) and Posterior Mean (PM) estimators are intractable. Hence, we propose to approximate the true posterior by free-form separable distribution using a gradient-like variational Bayesian approach, which allows a joint update of the shape parameters of the approximating marginals. Applications on simulated and real datasets, obtained from Herschel space observatory, show a good quality of reconstruction. Furthermore, compared to conventional methods, our method gives a higher resolution while conserving photometery and reducing noise.