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

TL;DR

This paper introduces a method to simplify Jacobi sets in 2D scalar fields by adjusting the underlying data, effectively removing noise-induced components while preserving essential structures, with applications demonstrated on various datasets.

Contribution

The paper presents a novel data modification approach for simplifying Jacobi sets, addressing limitations of previous methods by controlling how function values change to reduce complexity.

Findings

Effective removal of noise-induced Jacobi set components

Preservation of key structural features in scalar fields

Comparable or improved performance over simple smoothing techniques

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.