SIAM Review, Volume 65, Issue 3, Page 601-685, August 2023.
Over the last 30 years a plethora of variational regularization models for image reconstruction have 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 [L. I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259--268], which relies on the minimization of the edge-preserving total variation (TV) seminorm as a regularization 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 space variance. Space-variant models can indeed overcome the intrinsic inability of TV to describe local features (strength, sharpness, directionality) by means of an adaptive mathematical modeling which accommodates local regularization weighting, variable smoothness, and anisotropy. Those ideas can further be cast in the flexible Bayesian framework of generalized Gaussian distributions and combined with maximum likelihood and hierarchical optimization approaches for efficient hyperparameter 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.

中文翻译：

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成像中的全变差正则化及其之外：空间方差的作用

《SIAM 评论》，第 65 卷，第 3 期，第 601-685 页，2023 年 8 月。
在过去的 30 年里，应用数学界提出并彻底检验了大量用于图像重建的变分正则化模型。其中，数学图像处理基础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 [LI Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259--268]，它依赖于边缘保留总变差 (TV) 半范数的最小化作为正则化项。尽管其（通常是有限的）简单性，但由于其透彻的分析和数值理解，该模型仍然在许多应用中得到广泛采用，并用作评估现代基于学习的图像重建方法性能的基准。在多年来提出的许多电视扩展中，大类基于空间方差的概念。空变模型确实可以通过适应局部正则化权重、可变平滑度和各向异性的自适应数学模型来克服电视固有的无法描述局部特征（强度、清晰度、方向性）的问题。这些想法可以进一步融入广义高斯分布的灵活贝叶斯框架中，并与最大似然和分层优化方法相结合，以实现有效的超参数估计。在这项工作中，我们回顾并联系了空变电视型图像重建模型领域的主要贡献，特别关注其贝叶斯解释，这为新的令人兴奋和未经探索的研究方向铺平了道路。