This article proposes to integrate two Reeb graphs with the information of their isosurfaces' inclusion relation. As computing power evolves, there arise numerical data that have small-scale physics inside larger ones -- for example, small clouds in a simulation can be contained inside an atmospheric layer, which is further contained in an enormous hurricane. Extracting such inclusions between isosurfaces is a challenge for isosurfacing: the user would have to explore the vast combinations of isosurfaces $(f_1^{-1}(l_1), f_2^{-1}(l_2))$ from scalar fields $f_i: M(n) \to \mathbb{R}$, $i = 1, 2$, where $M$ is an $n$-dimensional domain manifold and $f_i$ are physical quantities, to find inclusion of one isosurface within another. For this, we propose the \textit{Reeb complement}, a topological space that integrates two Reeb graphs with the inclusion relation. The Reeb complement has a natural partition that classifies equivalent containment of isosurfaces. This is a handy characteristic that lets the Reeb complement serve as an overview of the inclusion relationship in the data. We also propose level-of-detail control of the inclusions through simplification of the Reeb complement. We demonstrate that the relationship of two independent scalar fields can be extracted by taking the product of Reeb graphs (which we call the Reeb product) and by then subtracting the projection of the Reeb space, which opens up a new possibility for feature analysis.

翻译：本文提出将两个Reeb图与其等值面包含关系信息进行整合的方法。随着计算能力的演进，数值数据中开始出现小尺度物理现象嵌套于大尺度结构中的情况——例如模拟中的小尺度云团可能包含于大气层内，而大气层又进一步包含于巨型飓风之中。提取此类等值面间的包含关系对等值面分析构成挑战：用户需要从标量场 $f_i: M(n) \to \mathbb{R}$（其中 $i = 1, 2$，$M$ 为 $n$ 维定义域流形，$f_i$ 表示物理量）生成的海量等值面组合 $(f_1^{-1}(l_1), f_2^{-1}(l_2))$ 中探索一个等值面包含于另一个等值面的情况。为此，我们提出 \textit{Reeb补集}——一种整合了两个Reeb图及其包含关系的拓扑空间。该Reeb补集具有自然划分结构，可对等值面的等价包含关系进行分类。这一特性使得Reeb补集能够作为数据包含关系的概览图。我们还提出通过简化Reeb补集实现包含关系的多细节层次控制。实验表明，两个独立标量场的关系可通过Reeb图乘积（称为Reeb积）再减去Reeb空间投影来提取，这为特征分析开辟了新的可能性。