Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields

TL;DR

This paper introduces a method to simplify the Jacobi set of time-varying scalar fields, enabling clearer visualization and tracking of topological features over time.

Contribution

It presents a novel approach using robustness measures to compute a reduced Jacobi set for dynamic scalar fields, with mathematical analysis and implementation for 2D cases.

Findings

Effective simplification of Jacobi sets demonstrated on synthetic data.

Improved tracking of topological features in real-world datasets.

Method enhances clarity in visualizations of time-varying scalar fields.

Abstract

The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features.