Topology-Preserving Meshing of Implicit Scalar Fields via Monotonicity Constraints

2026-08-12

AI Summary

Existing approaches to extracting topological structures—such as Morse–Smale complexes—from implicit two-dimensional scalar fields rely on explicit field representations, which limits their applicability. This work proposes a topology-preserving meshing method that requires only pointwise evaluations of the scalar field. By constructing a piecewise-linear (PL) triangulation and enforcing monotonicity of mesh edges with respect to the scalar field, the method accurately recovers critical points and their connectivity without access to an explicit field representation. An adaptive refinement strategy, incorporating monotonicity violation detection and correction, ensures topological correctness while significantly enhancing the geometric fidelity of separatrices. Experimental results demonstrate that the approach reliably reconstructs the Morse–Smale complex and achieves high-quality geometric approximation through targeted 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.

Problem

• Morse-Smale complex
• implicit scalar field
• topology-preserving meshing
• critical points
• monotonicity constraints

Innovation

• monotonicity constraints
• implicit scalar fields
• Morse-Smale complex
• topology-preserving meshing
• piecewise-linear approximation