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 highdimensional 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.

Reliable detection and separation of components for solid objects defined with scalar fields

“…The component analysis is one of the problems in the scope of the general topological analysis including the evaluation of the genus and the number of disjoint components, critical points detection [12] and the Reeb graph construction [11]. While our method supports detection of the number of disjoint components, it is not directly applicable to other topological analysis problems.…”

“…Another exact analysis of the topology and geometry of the models defined in the implicit form was done in [10], where the class of models was limited to those defined by the primitive polynomials. Also the analysis of the discrete scalar field by using Reeb graphs along with the Morse theory was presented in [11] Other works on topological analysis of isosurfaces of scalar fields are discussed in [12].…”

A Survey of Topology‐based Methods in Visualization

“…The Morse crystals (maximal cells of the MS complex) are created collecting vertices with the same label pair. Because these approximated Morse crystals can be non‐simply connected, Harvey et al [HRP*12] use the Reeb graph of such crystals to find and close spurious loops.…”

Functorial hierarchical clustering with overlaps

“…Recently, with the emergence of the new field of topological data analysis (TDA), work has been done on topologically-based clustering methods. This includes the Mapper algorithm by Singh, Mémoli, and Carlsson [39], as well as work on persistence-based methods [11] and Reeb graphs [28].…”