  We study active restoration of noise-corrupted images generated via the Gibbs
probability of an Ising ferromagnet in external magnetic field. Ferromagnetism
accounts for the prior expectation of data smoothness, i.e. a positive
correlation between neighbouring pixels (Ising spins), while the magnetic field
refers to the bias. The restoration is actively supervised by requesting the
true values of certain pixels after a noisy observation. This additional
information improves restoration of other pixels. The optimal strategy of
active inference is not known for realistic (two-dimensional) images. We
determine this strategy for the mean-field version of the model and show that
it amounts to supervising the values of spins (pixels) that do not agree with
the sign of the average magnetization. The strategy leads to a transparent
analytical expression for the minimal Bayesian risk, and shows that there is a
maximal number of pixels beyond of which the supervision is useless. We show
numerically that this strategy applies for two-dimensional images away from the
critical regime. Within this regime the strategy is outperformed by its local
(adaptive) version, which supervises pixels that do not agree with their
Bayesian estimate. We show on transparent examples how active supervising can
be essential in recovering noise-corrupted images and advocate for a wider
usage of active methods in image restoration.
