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

Dhruv Meduri, Mohit Sharma, Vijay Natarajan

TL;DR

This work addresses clutter in Jacobi sets derived from time-varying scalar fields by introducing a robustness-based direct Jacobi-set simplification. It adapts the notion of stability from vector fields to gradient fields, using δ-sublevel sets of the gradient magnitude to cluster critical points and propagate tracks over time, yielding a simplified tracking graph $\mathbb{J}^*$. A mathematical analysis guarantees the existence of a corresponding simplified vector field under δ-perturbations, and a 2D implementation demonstrates substantial clutter reduction while preserving major feature tracks across synthetic and real-world datasets. The approach improves interpretability and reliability of feature tracking in spatiotemporal multivariate data and complements existing persistence-based methods.

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.

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

TL;DR

This work addresses clutter in Jacobi sets derived from time-varying scalar fields by introducing a robustness-based direct Jacobi-set simplification. It adapts the notion of stability from vector fields to gradient fields, using δ-sublevel sets of the gradient magnitude to cluster critical points and propagate tracks over time, yielding a simplified tracking graph . A mathematical analysis guarantees the existence of a corresponding simplified vector field under δ-perturbations, and a 2D implementation demonstrates substantial clutter reduction while preserving major feature tracks across synthetic and real-world datasets. The approach improves interpretability and reliability of feature tracking in spatiotemporal multivariate data and complements existing persistence-based methods.

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.
Paper Structure (9 sections, 4 theorems, 6 equations, 11 figures)

This paper contains 9 sections, 4 theorems, 6 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Robustness-based simplification of the Jacobi set
  5. Algorithm
  6. Analysis
  7. Implementation
  8. Experimental results
  9. Conclusions

Key Result

theorem 1

Let $A \subset \mathbb{C}$ be a simply connected region which is not the whole plane and let $a \in A$. There exists a unique analytic function $R : A \rightarrow \mathbb{C}$ satisfying the following properties:

Figures (11)

  • Figure 1: Jacobi set of a 2D synthetic bivariate field. (a) Scalar field $f(x,y) = x^2 + y^2$ (b) Scalar field $g(x,y) = x(y-8)$ (c) Level sets of $f$ (green) and $g$ (red). The Jacobi set $\mathbf{J}(f,g)$, shown in black, passes through the points where gradients of $f$ and $g$ align.
  • Figure 2: Identifying a Jacobi edge. (Left) $h_{\lambda_e}(v_1) < h_{\lambda_e}(a)$ and $h_{\lambda_e}(v_2) > h_{\lambda_e}(a)$, which implies that $ab$ is a regular edge. (Middle) Both $h_{\lambda_e}(v_1)$ and $h_{\lambda_e}(v_2)$ are smaller than $h_{\lambda_e}(a)$, which implies that $ab$ is a maximum. (Right) Edge $ab$ is a minimum Jacobi edge.
  • Figure 3: Jacobi set of a time-varying scalar field. (a) A synthetic 2D time-varying scalar field obtained by applying incremental rotations over time. Select time steps within the range [1..299]. (b) All time steps of the scalar field. stacked together to show the spiral path followed by the two pairs of maxima. (c) The Jacobi set of the time-varying scalar field is noisy. It is difficult to identify the two primary data features due to clutter. (d) Edges of the Jacobi set filtered based on the scalar value. Edges with endpoints having scalar value less than $1$ are discarded, leaving only a few Jacobi edges that are in the spatial proximity of important features.
  • Figure 4: (a) A synthetic 2D vector field visualized using a LIC image and annotated with a subset of its critical points: sources (blue), sink (red), and saddles (green). Isocontours of the vector magnitude field are shown in different colors. The corresponding sublevel sets contain subsets of critical points. All contours corresponding to the sublevel set of a critical point may not be visible. Contours at smaller isovalues may lie close to the critical point and hence are occluded by the glyph used to show the critical point. (b) The merge tree captures the evolution of the connected components of the sublevel set components. Nodes are annotated with the degree. The degree of the leaf nodes is equal to their Poincaré index, +1 for sources/sinks and -1 for saddles. The degree of an internal node is equal to the sum of degrees of all leaf nodes within the subtree rooted at the node.
  • Figure 5: A simplified gradient vector field is constructed within the unit disk $\overline{\mathbb{D}}$ and then transferred to the region $\overline{X}$.
  • ...and 6 more figures

Theorems & Definitions (5)

  • theorem 1: Riemann Mapping Theorem conway2012functions
  • theorem 2: conway2012functions2
  • theorem 3
  • proof
  • theorem 4: chazal2011computing