AI 中文总结
本研究针对隐式标量场的MSC提取问题，提出基于单调性约束的网格生成方法，实验表明该方法可生成保临界点拓扑的网格，针对性细化还能提升MSC分界线几何保真度。
AI 中文摘要
标量场的拓扑分析会产生Morse-Smale复形（MSC）等结构，用于汇总多尺度的显著特征。现有MSC提取算法通常假设输入场为显式表示，如离散采样网格，但可视化领域的最新进展使隐式场表示得到普及，这些算法的假设不再成立。本研究解决从隐式定义的二维标量场中提取MSC的问题，提出一种构造三角剖分分段线性（PL）网格的方法，旨在保留底层隐式标量场的临界点。核心见解是：若所有边相对于底层场单调，则生成的PL近似在临界点上拓扑一致。基于此，引入缓解单调性违反的细化过程，该方法仅需逐点评估和适度网格细化，在实验中生成的PL网格在临界点上是正确的；最后，证明额外的针对性细化可提高MSC分界线的几何保真度。
英文摘要
Topological analysis of scalar fields yields structures such as the Morse-Smale complex (MSC) that summarize salient features across multiple scales. Existing MSC extraction algorithms typically assume an explicit representation of the input field, such as a discretely sampled mesh. However, recent advances in visualization have popularized implicit field representations, for which these assumptions no longer hold. In this work, we address the problem of extracting an MSC from an implicitly defined 2D scalar field. We present a method for constructing a triangulated piecewise-linear (PL) mesh that aims to preserve the critical points of an underlying implicit scalar field. Our central insight is that if all edges are monotonic with respect to the underlying field, then the resulting PL approximation is topologically consistent with respect to critical points. Based on this insight, we introduce a refinement procedure that mitigates monotonicity violations. Requiring only pointwise evaluations and modest mesh refinement, the approach produces PL meshes that are correct with regards to critical points in our experiments. Finally, we demonstrate that additional targeted refinement improves the geometric fidelity of MSC separatrices.
Comments4 pages + 1 page of references, 5 figures, accepted as short paper at IEEE Vis 2026
