Survey and Review

Abstract

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.