Topology-Preserving Meshing of Implicit Scalar Fields via Monotonicity Constraints

Authors

Tanner Finken (University of Arizona), Jixian Li (University of Utah), Bei Wang (University of Utah), Hanqi Guo (The Ohio State University), Joshua A Levine (University of Arizona)

Presentation

Session

Big Data, Bigger Physics

Time

Friday, Nov 13, 08:45 – 08:54 (US/Eastern) · session 08:00 – 09:30

Location

Hall America south

Keywords

Implicit scalar fields, Morse–Smale complexes, implicit neural representations, mesh refinement

Abstract

Topological analysis of scalar fields yields structures such as the Morse–Smale complex (MSC) that summarize salient features across multiple scales. Existing MSC extraction algorithms typically assume an explicit representation of the input field, such as a discretely sampled mesh. However, recent advances in visualization have popularized implicit field representations, for which these assumptions no longer hold. In this work, we address the problem of extracting an MSC from an implicitly defined 2D scalar field. We present a method for constructing a triangulated piecewise-linear (PL) mesh that aims to preserve the critical points of an underlying implicit scalar field. Our central insight is that if all edges are monotonic with respect to the underlying field, then the resulting PL approximation is topologically consistent with respect to critical points. Based on this insight, we introduce a refinement procedure that mitigates monotonicity violations. Requiring only pointwise evaluations and modest mesh refinement, the approach produces PL meshes that are correct with regards to critical points in our experiments. Finally, we demonstrate that additional targeted refinement improves the geometric fidelity of MSC separatrices.

For Practitioners

This paper will be of interest to practitioners who work with scalar field data and topological analysis, including visualization scientists, scientific computing researchers, simulation scientists, computational engineers, computer graphics practitioners, and data scientists who use implicit neural representations or other implicit field models. It is particularly relevant for those seeking to extract and analyze topological structures such as the Morse–Smale complex (MSC) from data that is not in a mesh-based format. Practitioners can apply the methods presented in this paper to perform topological feature extraction on implicitly defined scalar fields, including fields represented by neural networks, surrogate models, or other continuous implicit representations. The proposed approach enables existing piecewise-linear MSC extraction workflows to be used on implicit data by constructing a triangulated mesh that aims to preserve the critical point structure of the underlying field. This can help practitioners identify and analyze important features, segment scalar fields, and study multi-scale behavior without requiring an explicit discretization of the data beforehand. Additionally, it may help practitioners to improve training methods of implicit models with regards to critical points preserved.