GASP: A Gradient-Aware Shortest Path Algorithm for Boundary-Confined Visualization of 2-Manifold Reeb Graphs

TL;DR

This paper introduces GASP, a novel algorithm for visualizing Reeb graphs on 2-manifolds that respects boundary constraints, compactness, and gradient alignment, improving the fidelity of topological representations.

Contribution

GASP is the first algorithm to generate boundary-confined, gradient-aligned Reeb graph visualizations, addressing limitations of existing methods.

Findings

GASP produces more accurate Reeb graph visualizations.

The algorithm outperforms the geometric barycenter method in qualitative assessments.

Quantitative evaluations show improved topological fidelity.

Abstract

Reeb graphs are an important tool for abstracting and representing the topological structure of a function defined on a manifold. We have identified three properties for faithfully representing Reeb graphs in a visualization: they should be constrained to the boundary, compact, and aligned with the function gradient. Existing algorithms for drawing Reeb graphs are agnostic to or violate these properties. In this paper, we introduce an algorithm to generate Reeb graph visualizations, called GASP, that is cognizant of these properties, thereby producing visualizations that are more representative of the underlying data. To demonstrate the improvements, the resulting Reeb graphs are evaluated both qualitatively and quantitatively against the geometric barycenter algorithm, using its implementation available in the Topology ToolKit (TTK), a widely adopted tool for calculating and visualizing Reeb graphs.