A Gradient-like Variational Bayesian Approach for Unsupervised Extended Emission Map-making from Spire/herschel Data

In this work, we study the problem of extended source emission high resolution map-making, which is tackled as an inverse problem. Therefore, an unsupervised Bayesian framework is proposed for estimating sky maps and all the related hyperparameters. For the forward problem, a detailed physical model is introduced to describe different instrument effects like: optical transfer function, pointing process and temperature drifts. Several models for pointing and optical transfer functions are implemented and a Gaussian distribution is attributed to noise model. Since we are interested in extended emission, a Markovian field accounting for four closest neighbors is used as a sky prior. In our unsupervised approach, we write the joint posterior of the sky and all the hyperparameters (prior correlation and noise parameters) as a function of the likelihood and the different priors. Nevertheless, its expression is complicated and neither the joint maximum a posteriori (JMAP) nor the posterior mean (PM) have an explicit form. Therefore, we propose a new gradient like variational Bayesian approach to tackle the problem of posterior approximation. In order to accelerate the convergence, shaping parameters are updated simultaneously like in a gradient method. We applied our approach for unsupervised map-making of simulated data and real SPIRE/Herschel data. The results show good performance for our method in term of reconstruction quality and hyperparameters estimation and a gain in spatial resolution up to 3 times compared to conventional methods.