Enhanced topology-sensitive clustering by reeb graph shattering

Abstract

Scalar-valued functions are ubiquitous in scientific research. Analysis and visualization of scalar functions defined on low-dimensional and simple domains is a well-understood problem, but complications arise when the domain is high-dimensional or topologically complex. Topological analysis and Morse theory provide tools that are effective in distilling useful information from such difficult scalar functions. A recently proposed topological method for understanding high-dimensional scalar functions approximates the Morse-Smale complex of a scalar function using a fast and efficient clustering technique. The resulting clusters (the so-called Morse crystals) are each approximately monotone and are amenable to geometric summarization and dimensionality reduction. However, some Morse crystals may contain loops. This shortcoming can affect the quality of the analysis performed on each crystal, as regions of the domain with potentially disparate geometry are assigned to the same cluster. We propose to use the Reeb graph of each Morse crystal to detect and resolve certain classes of problematic clustering. This provides a simple and efficient enhancement to the previous Morse crystals clustering. We provide preliminary experimental results to demonstrate that our improved topology-sensitive clustering algorithm yields a more accurate and reliable description of the topology of the underlying scalar function.