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

TL;DR

This paper adapts Morse-Cerf theory to analyze the evolution of critical points in time-varying piecewise-linear scalar fields, introducing new topological descriptors, algorithms, and experimental validation.

Contribution

It extends Morse-Cerf theory to piecewise-linear functions, providing novel representations, algorithms, and measures for analyzing dynamic scalar fields.

Findings

Introduced vertex and Cerf diagrams for PL functions

Developed an algorithm to compute Cerf diagrams

Validated methods on time-varying scalar field data

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.