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.

翻译：在过去30年中,应用数学界提出并彻底检查了许许多多关于图像重建的变异常规化模型,应用数学界提出了这些模型,其中,在数学图像处理基本课程中经常教授和学习的开创性原型是著名的Rudin-Osher-Fatemi(ROF)模型,该模型依赖将边缘保护全变异(TV)半调(Risite{ROF})作为常规化术语的最小化。尽管该模型(通常有限)简单,但它在许多应用中仍然被大量使用,并被用作评估现代学习型图像重建方法绩效的基准,这要归功于其透彻的分析和数字理解。在多年来提出的电视的许多扩展中,一个大类是基于emph{空间差异的理念。空间变异模型确实可以克服电视本身无法在常规化时描述 & emph{ local} 特征( 坚固、 精度、 方向性) 的适应性数学模型, 适应本地的调整权重度、 平滑度和 氮质化的模型。这些想法可以进一步在高层次和主要图像化的模型化框架中以高压化法化方式进行对比。