Symmetric second-order tensors are fundamental in various scientific and engineering domains, as they can represent properties such as material stresses or diffusion processes in brain tissue. In recent years, several approaches have been introduced and improved to analyze these fields using topological features, such as degenerate tensor locations, i.e., the tensor has repeated eigenvalues, or normal surfaces. Traditionally, the identification of such features has been limited to single tensor fields. However, it has become common to create ensembles to account for uncertainties and variability in simulations and measurements. In this work, we explore novel methods for describing and visualizing degenerate tensor locations in 3D symmetric second-order tensor field ensembles. We base our considerations on the tensor mode and analyze its practicality in characterizing the uncertainty of degenerate tensor locations before proposing a variety of visualization strategies to effectively communicate degenerate tensor information. We demonstrate our techniques for synthetic and simulation data sets. The results indicate that the interplay of different descriptions for uncertainty can effectively convey information on degenerate tensor locations.

翻译：对称二阶张量在众多科学与工程领域中具有基础性地位，因其能够表征材料应力或脑组织扩散过程等物理属性。近年来，学界已提出并改进了若干基于拓扑特征（如退化张量位置——即具有重复特征值的张量——或法向曲面）分析此类张量场的方法。传统上，这类特征的识别仅限于单一张量场。然而，为反映仿真与测量中的不确定性和变异性，构建张量场集合已成为常见做法。本研究探索了描述与可视化三维对称二阶张量场集合中退化张量位置的新方法。我们以张量模态为理论基础，在提出多种可视化策略以有效传达退化张量信息前，系统分析了该模态在表征退化张量位置不确定性方面的实用性。我们通过合成数据集与仿真数据集验证了所提技术。结果表明，不同不确定性描述方式的协同作用能有效传递退化张量位置的信息。