Topological Simplifcation of Jacobi Sets for Piecewise-Linear Bivariate 2D Scalar Fields with Adjustment of the Underlying Data

TL;DR

The paper tackles noise-driven complexity in Jacobi sets for 2D bivariate piecewise-linear fields by proposing a data-adjustment approach that collapses region cells guided by a neighborhood graph. The core method computes Jacobi sets, partitions the domain into Jacobi-separated regions, weights these regions by triangle measures, and greedily collapses whole regions to remove noise-induced components while preserving essential structures; four neighborhood-graph variants are explored, with variant A performing best. Across synthetic and real datasets (Cylinder Flow, Tensile Bar, Hurricane Isabel), the Collapse Algorithm (especially CA variant A) significantly reduces the number of Jacobi-set components (often by over an order of magnitude) with competitive runtimes, while smoothing-based methods either distort important features or fail to remove small components. The work highlights the importance of the neighborhood-graph design, demonstrates strong practical gains in interpretability, and points to future work on automatic thresholding, 3D extensions, and gradual collapsing for symmetry-sensitive data.

Abstract

Jacobi sets are an important tool to study the relationship between functions. Defined as the set of all points where the function's gradients are linearly dependent, Jacobi sets extend the notion of critical point to multifields. In practice, Jacobi sets for piecewise-linear approximations of smooth functions can become very complex and large due to noise and numerical errors. Existing methods that simplify Jacobi sets exist, but either do not address how the functions' values have to change in order to have simpler Jacobi sets or remain purely theoretical. In this paper, we present a method that modifies 2D bivariate scalar fields such that Jacobi set components that are due to noise are removed, while preserving the essential structures of the fields. The method uses the Jacobi set to decompose the domain, stores the and weighs the resulting regions in a neighborhood graph, which is then used to determine which regions to join by collapsing the image of the region's cells. We investigate the influence of different tie-breaks when building the neighborhood graphs and the treatment of collapsed cells. We apply our algorithm to a range of datasets, both analytical and real-world and compare its performance to simple data smoothing.

Figures (8)

• Figure 1: Section of the cylinder flow dataset (a) is an example of applying the different neighborhood graphs. (b) classical neighborhood graph A, (c) neighborhood graph B with connected negative cells, (d) neighborhood graph C with connected positive cells, and (e) neighborhood graph A with connected cells corresponding to the neighborhood.
• Figure 2: Example of collapsing a cell in 1D. Left before collapsing the cell BC, right after collapsing. The dashed line shows when the neighborhood is not taken into account. The solid line shows when it is taken into account. The red color shows a positive and the blue a negative orientation.
• Figure 3: Example for the collapse of a triangle cell in the range for all 4 cases to be considered: V1 shows the collapse to a point, and V2 shows the collapse to a line. The green dashed edges connect the cell with a cell to be collapsed. The orange dashed lines are the edges to be scrambled.
• Figure 4: Comparison of the calculated Jacobi sets in the Cylinder Flow dataset for the Loop Subdivision (a), the Binomial filter (b), the Gaussian filter (c), the Collapse Algorithm variant B with $t = 0.0001$ (d), Collapse Algorithm variant C with $t = 0.0001$ (e), and Collapse Algorithm variant D with $t = 0.0001$ (f). Three regions of interest (ROI) are highlighted in color for visual comparison.
• Figure 5: The datasets from \ref{['fig:teaser']} are mapped into the range. The border of the domain is shown in green, the Jacobi sets in black, and the grid in orange for the original data (a) and the dataset after applying CA variant A (b) in orange.