Topological Simplifcation of Jacobi Sets for Piecewise-Linear Bivariate 2D Scalar Fields

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Abstract

Jacobi sets are an important method to investigate the relationship between Morse functions. The Jacobi set for two Morse functions is the set of all points where the functions' gradients are linearly dependent. Both the segmentation of the domain by Jacobi sets and the Jacobi sets themselves have proven to be useful tools in multi-field visualization, data analysis in various applications, and for accelerating extraction algorithms. On a triangulated grid, they can be calculated by a piecewise linear interpolation. In practice, Jacobi sets can become very complex and large due to noise and numerical errors. Some techniques for simplifying Jacobi sets exist, but these only reduce individual elements such as noise or are purely theoretical. These techniques often only change the visual representation of the Jacobi sets, but not the underlying data. In this paper, we present an algorithm that simplifies the Jacobi sets for 2D bivariate scalar fields and at the same time modifies the underlying bivariate scalar fields while preserving the essential structures of the fields. We use a neighborhood graph to select the areas to be reduced and collapse these cells individually. We investigate the influence of different neighborhood graphs and present an adaptation for the visualization of Jacobi sets that take the collapsed cells into account. We apply our algorithm to a range of analytical and real-world data sets and compare it with established methods that also simplify the underlying bivariate scalar fields.