Scalar functions are virtually ubiquitous in scientific research. A vast amount of research has been conducted in visualization and exploration of low-dimensional data during the last few decades, but adapting these techniques to high-dimensional, topologically-complex data remains challenging. Traditional metric-preserving dimensionality reduction techniques suffer when the intrinsic dimension of data is high, as the metric cannot generally survive projection into low dimensions. The metric distortion can be arbitrarily large, and preservation of topological structure is not guaranteed, resulting in a misleading view of the data.
When preservation of geometry is not possible, topological analysis provides a promising alternative. As an example, simplicial homology characterizes the structure of a topological space (i.e. a simplicial complex) via its intrinsic topological features of various dimensions. Unfortunately, this information can be abstract and difficult to comprehend. The ranks of these homology groups (the Betti numbers) offer a simpler, albeit coarse, interpretation as the number of voids of each dimension. In high dimensions, these approaches suffer from exponential time complexity, which can render them impractical for use with real data.
In light of these difficulties, we turn to an alternative type of topological characterization. We investigate the Reeb graph as a visualization and analysis tool for such complex data. The Reeb graph captures the topology of the set of level sets of a scalar function, providing a simple, intuitive, and informative topological representation. We present the first sub-quadratic expected time algorithm for computing the Reeb graph of an arbitrary simplicial complex, opening up the possibility of using the Reeb graph as a tool for understanding high-dimensional data.
While the Reeb graph effectively captures some topological structure, it is still somewhat terse. The Morse-Smale complex summarizes a scalar function by by partitioning the domain according to gradient flow source and destination. While this information is rich and useful, it can be impractical to compute in high dimensions. The Morse crystal decomposition approximates the Morse-Smale complex, accepting a significant gain in computational efficiency for potential under-segmentation of the domain. We demonstrate how the novel Reeb graph algorithm enhances the Morse crystal decomposition by potentially reducing the degree of under-segmentation that occurs.
A Reeb graph without loops is known as a contour tree. Contour trees can arise, for example, when the domain is simply connected. In their raw forms, Reeb graphs and contour trees can be difficult to interpret, as their graph-like structures can be complex and hard to visualize. We propose a method for visualizing a high-dimensional scalar function by constructing an ensemble of two-dimensional terrain models which share its contour tree. These "topological landscapes" perfectly preserve certain metric information, providing intuition about the topology and geometry of the scalar function.
From these foundations, we develop a collaborative visual analytics suite for protein folding and computational chemistry researchers. This software focuses on the topological landscape, allowing users to interactively visualize and explore massive high-dimensional molecular dynamics datasets. Users collaborate in annotating interesting biochemical events and assemble them into cohesive narratives using a simple storyboard interface.