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Computing Reeb Graphs as a Union of Contour Trees.

Harish Doraiswamy, Vijay Natarajan

IEEE Transactions on Visualization and Computer Graphics

|April 25, 2012

Summary

This paper presents a fast algorithm for computing the Reeb graph of piecewise-linear functions. By leveraging contour trees and union-find operations, it significantly speeds up topological analysis for large datasets.

Area of Science:

Topological Data Analysis
Computer Graphics
Scientific Visualization

Background:

The Reeb graph is crucial for understanding the topology of scalar functions by tracking level set evolution.
Existing methods for Reeb graph computation can be slow, especially for complex or large datasets.
Piecewise-linear (PL) functions defined over manifolds and non-manifolds present computational challenges.

Purpose of the Study:

To develop a fast and efficient algorithm for computing the Reeb graph of piecewise-linear scalar functions.
To enable topological analysis on large datasets that may not fit into memory.
To improve upon existing generic Reeb graph computation algorithms.

Main Methods:

The algorithm leverages the contour tree algorithm for efficient Reeb graph computation.

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It divides the input data into subvolumes with loop-free Reeb graphs using the scalar function's join tree.

Reeb graphs are computed by combining contour trees of subvolumes, utilizing union-find operations.

Main Results:

The proposed algorithm significantly outperforms current generic methods, achieving speedups of up to two orders of magnitude.
Performance is comparable to specialized algorithms designed for restricted input types.
The method effectively handles large-scale data that exceeds available memory.

Conclusions:

The developed algorithm provides a fast and scalable solution for Reeb graph computation.
It offers a practical approach for topological analysis in scientific visualization and data analysis.
The method's efficiency and ability to handle large data open new possibilities for complex data exploration.