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 $f_i: M \to \mathbb{R}$, $i = 1, 2$, where $M$ is a 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 to let 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.

翻译：本文提出将两个Reeb图与它们的等值面包含关系信息进行整合。随着计算能力的发展，数值数据中出现了大尺度物理现象包含小尺度结构的情况——例如，模拟中的小云团可能被包裹在大气层中，而大气层又进一步被包含于巨型飓风内。提取等值面间的此类包含关系对等值面技术构成挑战：用户需要遍历标量场$f_i: M \to \mathbb{R}$（$i = 1, 2$，其中$M$为定义域流形，$f_i$为物理量）的等值面组合$(f_1^{-1}(l_1), f_2^{-1}(l_2))$，以发现一个等值面被另一个包含的情况。为此，我们提出\textit{Reeb补集}——一种集成了两个Reeb图及其包含关系的拓扑空间。该空间具有自然划分，可对等值面的等价包含关系进行分类。这一特性使得Reeb补集能够便捷地作为数据中包含关系的全局概览。我们还通过简化Reeb补集实现了包含关系的细节层次控制。