Reeb Complements for Exploring Inclusions Between Isosurfaces From Two Scalar Fields

TL;DR

It is demonstrated that the relationship of two independent scalar fields can be extracted by taking the product of Reeb graphs and by then subtracting the projection of the Reeb space, which opens up a new possibility for feature analysis.

Abstract

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.

Figures (12)

• Figure 1: The Reeb graph summarizes the landscape of a height field $f: M \to \mathbb{R}$. Each contour, i.e. a connected component of a preimage of $f$, becomes a point in this 1-dimensional topological space illustrated in this figure.
• Figure 2: Example where the contour topologically changes but the number of the contours stays to be one.
• Figure 3: Two isosurfaces without intersection, in a Euclidean space.
• Figure 4: Topological change of inclusion relationship.
• Figure 5: Contour lines of $f_1$ and the corresponding Reeb graph.

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Definition 3
• Definition 4