Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

翻译：拓扑抽象提供了一种概括向量场行为的方法，但由于数值精度问题，稳健地计算这些抽象可能具有挑战性。一种替代方案是采用离散方法表示向量场，该方法在输入网格中构建满足福尔曼离散莫尔斯理论引入的准则的单纯形对集合。尽管在标量场梯度的受限情况下存在多种计算配对的方法，但对于向量场的一般情况，最先进的算法需要昂贵的优化过程。本文针对不随时间变化的二维三角化向量场，提出了一种快速、新颖的单纯形配对方法。我们方法的关键见解在于，可以借鉴构建离散梯度场时采用的局部评估策略，即网格中的每个单纯形仅由其一个顶点进行考量。具体而言，我们观察到对于输入网格中的任何边，可以唯一地分配一个外向流动方向。我们可以进一步在每个顶点扩展这种一致的外向流动概念，这对应于标量场情况下的下坡流概念。利用外向流动概念，我们设计了一种线性时间算法，该算法逐个处理每个顶点的（外向）邻域，类似于标量场采用的方法。我们将构建离散向量场的方法与提取、简化和可视化拓扑特征的方法相结合。在解析数据和模拟数据上的实证结果表明，该方法在运行时间上实现了显著提升，产生的特征与当前最先进技术相似，并展示了简化方法在大型复杂流场中的应用。