SIAM Review, Volume 65, Issue 3, Page 599-599, August 2023.
Apart from a short erratum, which concerns the correction of some coefficients in a differential equation in the original paper, this issue contains two Survey and Review articles. “On and Beyond Total Variation Regularization in Imaging: The Role of Space Variance,” authored by Monica Pragliola, Luca Calatroni, Alessandro Lanza, and Fiorella Sgallari, reviews total variation (TV)-type image reconstruction algorithms with a focus on Bayesian interpretations. The paper scientifically travels across various disciplines by considering a standard example problem to highlight extensions for the TV regularization model. A main contribution is a space-variant framework which allows one to describe the contents of an image at a local scale. Important applications of space-variant models are tomography, e.g., magnetic resonance imaging, electrical impedance tomography, positron emission tomography, and photoacoustic tomography, or noninvasive digital reconstruction, e.g., for ancient frescoes, illuminated manuscripts, surface colorization, etc. The unified view of many of the different models within the Bayesian framework enables one to design flexible and adaptive image regularization functionals which take advantage of the form of the underlying gradient distributions through statistical approaches. The paper contains theoretical results as well as sections on algorithmic optimization (based on the alternating direction methods of multipliers) and numerical tests for examples from image deblurring. Thus it should be interesting for researchers from several disciplines. “What Are Higher-Order Networks” is a question raised and answered by Christian Bick, Elizabeth Gross, Heather A. Harrington, and Michael T. Schaub. In short, higher-order networks are a refurbishment of graphs, removing/overcoming some of the limitations of pairwise relationships by enabling the modeling of polyadic relations in real-world systems, such as reactions in biochemical systems with several species or reagents, or interactions of multiple people in social networks. The main topics of discussion are the understanding of the “shape” of data (by identifying and classifying topological and geometrical properties of the data), the modeling of relational data via higher-order networks, and network dynamical systems (describing couplings between dynamical units). The focus of the presentation is on the mathematical aspects of the topics, but a multitude of applications are mentioned. The impressive list of references comprises 316 entries. We believe the paper to be interesting for a broad audience.

中文翻译：

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调查与回顾

《SIAM 评论》，第 65 卷，第 3 期，第 599-599 页，2023 年 8 月。
除了涉及原论文中微分方程中某些系数的修正的简短勘误之外，本期还包含两篇调查和评论文章。Monica Pragliola、Luca Calatroni、Alessandro Lanza 和 Fiorella Sgallari 撰写的《成像中的全变分正则化及其之外：空间方差的作用》回顾了全变分 (TV) 型图像重建算法，重点关注贝叶斯解释。该论文通过考虑一个标准示例问题来科学地跨越各个学科，以强调电视正则化模型的扩展。一个主要的贡献是一个空间变化的框架，它允许人们在局部尺度上描述图像的内容。空变模型的重要应用是断层扫描，例如磁共振成像，电阻抗断层扫描、正电子发射断层扫描和光声断层扫描，或无创数字重建，例如古代壁画、彩绘手稿、表面着色等。贝叶斯框架内许多不同模型的统一视图使人们能够设计灵活且自适应图像正则化函数，通过统计方法利用底层梯度分布的形式。本文包含理论结果以及算法优化部分（基于乘法器的交替方向方法）以及图像去模糊示例的数值测试。因此，来自多个学科的研究人员应该会对此感兴趣。“什么是高阶网络”是 Christian Bick、Elizabeth Gross 提出并回答的问题，希瑟·A·哈林顿 (Heather A. Harrington) 和迈克尔·T·绍布 (Michael T. Schaub)。简而言之，高阶网络是图的翻新，通过在现实世界系统中对多元关系进行建模，消除/克服成对关系的一些限制，例如生化系统中与多种物种或试剂的反应，或相互作用社交网络中的多人。讨论的主要主题是对数据“形状”的理解（通过识别和分类数据的拓扑和几何属性）、通过高阶网络对关系数据进行建模以及网络动态系统（描述动态单元之间的耦合） ）。演示的重点是主题的数学方面，但也提到了许多应用。令人印象深刻的参考文献列表包含 316 个条目。