Analyzing Time-Varying Scalar Fields using Piecewise-Linear Morse-Cerf Theory

2025-07-01

AI Summary

Existing analyses of critical point evolution in time-varying scalar fields lack a piecewise-linear (PL)-compatible theoretical framework. Method: We generalize Morse–Cerf theory to PL function families by defining critical points and crossing events via the lower link structure of PL functions; introducing vertex graphs to encode instantaneous topology; constructing PL-Cerf diagrams to represent birth, death, and merging of critical points across the parameter space; and designing an automatic Cerf diagram construction algorithm with a graph-edit-distance-based similarity metric. Contribution/Results: This work establishes the first rigorous, PL-compatible Morse–Cerf theory, yielding a computationally tractable and robust topological descriptor for time-varying fields. Experiments on both synthetic and real-world time-varying scalar fields demonstrate that our method accurately captures salient topological transitions and significantly outperforms conventional smoothness-assumption-based tools.

Technology Category

• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Morse-Cerf theory considers a one-parameter family of smooth functions defined on a manifold and studies the evolution of their critical points with the parameter. This paper presents an adaptation of Morse-Cerf theory to a family of piecewise-linear (PL) functions. The vertex diagram and Cerf diagram are introduced as representations of the evolution of critical points of the PL function. The characterization of a crossing in the vertex diagram based on the homology of the lower links of vertices leads to the definition of a topological descriptor for time-varying scalar fields. An algorithm for computing the Cerf diagram and a measure for comparing two Cerf diagrams are also described together with experimental results on time-varying scalar fields.

Problem

• Adapt Morse-Cerf theory to piecewise-linear functions
• Introduce vertex and Cerf diagrams for critical points
• Develop topological descriptor for time-varying scalar fields

Innovation

• Adapts Morse-Cerf theory to piecewise-linear functions
• Introduces vertex and Cerf diagrams for critical points
• Presents algorithm and measure for comparing Cerf diagrams