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Over the last 30 years a plethora of variational regularisation models forimage reconstruction has been proposed and thoroughly inspected by the appliedmathematics community. Among them, the pioneering prototype often taught andlearned in basic courses in mathematical image processing is the celebratedRudin-Osher-Fatemi (ROF) model \cite{ROF} which relies on the minimisation ofthe edge-preserving Total Variation (TV) semi-norm as regularisation term.Despite its (often limiting) simplicity, this model is still very much employedin many applications and used as a benchmark for assessing the performance ofmodern learning-based image reconstruction approaches, thanks to its thoroughanalytical and numerical understanding. Among the many extensions to TVproposed over the years, a large class is based on the concept of \emph{spacevariance}. Space-variant models can indeed overcome the intrinsic inability ofTV to describe \emph{local} features (strength, sharpness, directionality) bymeans of an adaptive mathematical modelling which accommodates localregularisation weighting, variable smoothness and anisotropy. Those ideas canfurther be cast in the flexible Bayesian framework of generalised Gaussiandistributions and combined with maximum likelihood and hierarchicaloptimisation approaches for efficient hyper-parameter estimation. In this work,we review and connect the major contributions in the field of space-variantTV-type image reconstruction models, focusing, in particular, on their Bayesianinterpretation which paves the way to new exciting and unexplored researchdirections.

On and beyond Total Variation regularisation in imaging: the role of space variance