We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.

Fig. 1. Application of DMD and constrained DMD to an artificial time series that consists of four different patterns: linear trend, two seasonal patterns with the periods 7 and 28, and noise. Thus, the superposed time series is a typical example of daily data exhibiting weekly and monthly patterns. While DMD detects the weekly pattern with an identified period of 6.90, it fails to compute the correct trend and monthly pattern. Incorporating both frequencies, our constrained DMD identifies all patterns correctly.

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Figure 1: Comparison of the original connectivity construction for a local bilinear Jacobi set (left) and our novel reduced connectivity (right). The color coding (blue-to-red) shows the gradient alignment field of two analytic functions, and the extracted/computed Jacobi set is displayed by solid black lines. The center images show zoomed-in details of two regions in the visualizations (marked by white outlines). It can be observed that the reduced connectivity construction results in fewer edges and, therefore, in a clearer visual representation.