Conforming Morse-Smale Complexes

Abstract

Morse-Smale (MS) complexes have been gaining popularity as a tool for feature-driven data analysis and visualization. However, the quality of their geometric embedding and the sole dependence on the input scalar field data can limit their applicability when expressing application-dependent features. In this paper we introduce a new combinatorial technique to compute an MS complex that conforms to both an input scalar field and an additional, prior segmentation of the domain. The segmentation constrains the MS complex computation guaranteeing that boundaries in the segmentation are captured as separatrices of the MS complex. We demonstrate the utility and versatility of our approach with two applications. First, we use streamline integration to determine numerically computed basins/mountains and use the resulting segmentation as an input to our algorithm. This strategy enables the incorporation of prior flow path knowledge, effectively resulting in an MS complex that is as geometrically accurate as the employed numerical integration. Our second use case is motivated by the observation that often the data itself does not explicitly contain features known to be present by a domain expert. We introduce edit operations for MS complexes so that a user can directly modify their features while maintaining all the advantages of a robust topology-based representation.

Critical Points to Determine Persistence Homology

“…A future direction is to consider the adaptability of the proposed approach to discrete MS complexes, which can be computed efficiently [29]- [35].…”

TopoMS: Comprehensive topological exploration for molecular and condensed‐matter systems

“…While the authors presented a solution, the algorithm required a serial traversal, and is too slow for practical use for QTAIM analysis. Instead, in T opo MS , we utilize an approach that allows fast parallel computation of discrete gradient fields that also conforms to a prior labeling computed through numeric integration …”

“…The boundary map is nonzero only for cells having vertices with different labels in . We use the algorithm proposed by Gyulassy et al In the following discussion, we denote a cell that has been identified as critical by pairing it with itself, e.g., . Furthermore, a cell is assigned if and only if it has been identified as critical or paired in a discrete gradient vector.…”

Topologically Controlled Lossy Compression

“…1). In many applications, persistence diagrams help users as a visual guide to interactively tune simplification thresholds in topology-based, multi-scale data segmentation tasks based on the Reeb graph [9,19,34,42,45] or the Morse-Smale complex [21,22]. Distance: In order to evaluate the quality of compression algorithms, several metrics have been defined to evaluate the distance between the decompressed data, noted g : M → R, and the input data, f :…”