Unification of some deterministic and probabilistic methods for the solution of linear inverse problems via the principle of maximum entropy on the mean

This paper deals with the resolution of linear inverse problems. We present a mathematical study of the principle of maximum entropy on the mean (PMEM), and show that it is possible to derive from this principle many deterministic and probabilistic regularization techniques, including the well known Tikhonov method as well as the classical entropy method. In particular, within the deterministic family, full attention is devoted to the regularization principle of WIPE, a methodology recently introduced in radio imaging and optical interferometry. The principle in question is based on the concept of resolution as it is usually introduced in physics. The infinite-dimensional linearly constrained optimization problem underlying the PMEM is solved by means of a dual strategy: recent developments on partially finite convex programming are applied to our specific problem. To illustrate our analysis, we also present a few numerical simulations, in which the regularizer of WIPE is compared with entropy.