The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features.

翻译：双变量标量场的雅可比集是指两个组成标量场的梯度彼此对齐的点集。它刻画了双变量场中拓扑结构发生变化的区域。雅可比集是临界点的双变量类比，可能对应着感兴趣的特征。在时变场的特定情形下，当其中一个标量场为时间变量时，雅可比集对应于临界点的时间轨迹，并可作为特征追踪图。双变量场或时变标量场的雅可比集通常较为复杂，会导致可视化结果杂乱而难以分析。本文致力于解决雅可比集的简化问题。具体而言，我们使用时变标量场场景提出一种计算简化雅可比集的方法。该方法基于称为鲁棒性的稳定性度量——该度量最初为向量场开发，有助于捕捉临界点的结构稳定性。我们还对该方法进行了数学分析，并描述了针对二维时变标量场的实现方案。在合成数据集和真实数据集上的应用验证了该方法在特征追踪方面的有效性。