  Over the last 30 years a plethora of variational regularisation models for
image reconstruction has been proposed and thoroughly inspected by the applied
mathematics community. Among them, the pioneering prototype often taught and
learned in basic courses in mathematical image processing is the celebrated
Rudin-Osher-Fatemi (ROF) model \cite{ROF} which relies on the minimisation of
the edge-preserving Total Variation (TV) semi-norm as regularisation term.
Despite its (often limiting) simplicity, this model is still very much employed
in many applications and used as a benchmark for assessing the performance of
modern learning-based image reconstruction approaches, thanks to its thorough
analytical and numerical understanding. Among the many extensions to TV
proposed over the years, a large class is based on the concept of \emph{space
variance}. Space-variant models can indeed overcome the intrinsic inability of
TV to describe \emph{local} features (strength, sharpness, directionality) by
means of an adaptive mathematical modelling which accommodates local
regularisation weighting, variable smoothness and anisotropy. Those ideas can
further be cast in the flexible Bayesian framework of generalised Gaussian
distributions and combined with maximum likelihood and hierarchical
optimisation approaches for efficient hyper-parameter estimation. In this work,
we review and connect the major contributions in the field of space-variant
TV-type image reconstruction models, focusing, in particular, on their Bayesian
interpretation which paves the way to new exciting and unexplored research
directions.
