AI 中文总结
本文研究双变量场的特质诱导合并树（TIMT）精确计算，构建与精确合并树同构的加权图，建立顶点采样插值误差上界，用CGAL和VTK实现并在数据集上验证。
AI 中文摘要
特质诱导合并树（TIMT）是一种基于拓扑的稳健方法，通过分析属性空间中用户指定特质诱导的距离场，用于选择和浏览多变量数据中的特征水平集。现有TIMT计算通常在网格顶点处对该距离场进行采样，并假设其为分段线性插值，但原始网格上的欧氏距离-特质函数通常并非分段线性。因此，所得合并树可能遗漏零值特征，还可能扰动极小值及合并事件的位置与数值。我们研究分段线性双变量场的TIMT精确计算，首先聚焦于点特质。我们证明，每个四面体内部的受限子水平集是凸的，因此具有平凡的局部合并树结构，这意味着全局拓扑变化仅通过单纯形边界的粘合产生。基于此观察，我们构建一个加权图，其合并树与诱导距离场的精确合并树同构。我们进一步将TIMT与雅可比集关联，展示TIMT的非零边事件如何由基础双变量映射的奇异结构定位。我们建立顶点采样线性插值误差的理论上界，该上界以范围内投影网格边的最大长度表示。我们讨论向线、线段和有限点集特质的扩展，并使用CGAL和VTK稳健实现该方法，在合成及真实世界数据集上展示结果。
英文摘要
Trait-induced merge trees (TIMTs) provide a robust topology-based method for selecting and browsing feature level sets in multivariate data by analyzing the distance field induced by a user-specified trait in attribute space. Existing TIMT computations typically sample this distance field at mesh vertices and assume piecewise-linear interpolation, although the Euclidean distance-to-trait function is generally not piecewise linear on the original mesh. As a result, the resulting merge tree may miss zero-valued features and may perturb the locations and values of minima and merge events. We study the exact computation of TIMTs for piecewise-linear bivariate fields, focusing first on point traits. We show that the restricted sublevel sets inside each tetrahedron are convex and therefore have trivial local merge-tree structure, implying that global topological changes arise only through gluing across simplex boundaries. Based on this observation, we construct a weighted graph whose merge tree is isomorphic to the exact merge tree of the induced distance field. We further relate TIMTs to Jacobi sets, showing how nonzero edge events of the TIMT are localized by the singular structure of the underlying bivariate map. We establish a theoretical upper bound on the error of the vertex-sampled linear interpolation, expressed in terms of the maximum length of projected mesh edges in the range. We discuss extensions to line, line-segment, and finite point-set traits, and implement the method robustly using CGAL and VTK, demonstrating results on both synthetic and real-world datasets.
Comments12 pages, 10 figures, To appear in an IEEE VGTC workshop "TopoInVis Connect 2026 - Topology meets Artificial Intelligence"
