Localized Evaluation for Constructing Discrete Vector Fields

TL;DR

The paper addresses robust, scalable topological representation of 2D steady vector fields by introducing a local outward-star based method to construct discrete vector fields. This enables linear-time processing via a LowerStar-inspired homotopy expansion and yields a discrete representation suitable for extracting, simplifying, and visualizing topological features. Empirical results show dramatic runtime improvements over global optimization-based methods (e.g., FastCVT) while preserving key topological structures, and demonstrate effective simplification of large, complex flows. The approach is modular, parallelizable, and extensible to integration with visualization toolchains, enabling scalable analysis of real-world flow data such as ocean currents.

Abstract

Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

Figures (9)

• Figure 1: We show three possible configurations for which $f(v_0,v_1)>0$ for a simplex $\sigma = \{v_0,v_1\}$, resulting in outward flow from $v_0$ (indicated by the blue ellipse. Dotted lines indicate how $F(\sigma)$ and the dot product for $f(v_0,v_1)$ are computed. In (a) both vectors are pointing away from $v_0$. (b) The vectors at $v_0$ and $v_1$ both flow towards $\sigma$, but $F(v_0)$ dominates. (c) Neither vector flows towards $\sigma$, however since $f(v_0,v_1)>0$ we define the flow as outward from $v_0$.
• Figure 2: Examples of outward stars for a vertex $x$, with blue indicating outward flow. (a) $x$ has 3 edges (associated with $v_0,v_1,v_2$) and thus also includes two triangles in $\mathrm{St}^\leftrightarrow(x)$. (b) No edges are in $\mathrm{St}^\leftrightarrow(x)$. (c) Four non-adjacent edges are included in $\mathrm{St}^\leftrightarrow(x)$, resulting in two triangles in $\mathrm{St}^\leftrightarrow(x)$ (involving $v_1, v_2$ and $v_4, v_5$). (d) All surrounding edges and triangles are included in $\mathrm{St}^\leftrightarrow(x)$ (e) A lone edge is in $\mathrm{St}^\leftrightarrow(x)$, involving $v_4$. We also show edges assigned to $\mathrm{St}^\leftrightarrow(v_3)$ (pink) and $\mathrm{St}^\leftrightarrow(v_4)$ (purple). In this case, the triangle will not be assigned to any outward star.
• Figure 3: We show two different examples of saddle simplification. (a) in our input configuration, $\sigma$ has an index 0 separatrix that arrives at $x$ and an index 1 separatrix that arrives from $\tau$ (only two of the four possible separatrices are shown). (b) Cancelling the index 0 separatrix results in removing both $x$ and $\sigma$. (c) Cancelling the index 1 separatrix results in removing $\sigma$ and $\tau$.
• Figure 4: Saddle-orbit cancellation. In (a) we show a saddle $\sigma$ whose index 0 separatrixes both terminate in orbits. Note for size we show an orbit on a single triangle, but a similar situation could occur with a longer closed $V$-path. (b) Reversing the $V$-path that follows the orbit around $\alpha$ results in sliding $\sigma$ to $\sigma'$ and removing the orbit. (c) If one tries to reverse the $V$-path that orbits $\tau$, $\sigma'$ will slide to $\sigma"$ and in doing so the orbit around $\tau$ will be removed, but the orbit around $\alpha$ will reappear.
• Figure 5: Comparison of Vector Field produced by Formula \ref{['eq:vectorField']} (a) Continuous evaluation using (Colors: Cyan=Attracting Focus, White=Saddle, Dark-Grey=Center,Orange=Repelling Node) (b) Discrete Vector Field critical points produced by our algorithm (Discrete Sphere Colors: Red=Index 2 CP, White=Index 1 CP, Blue=Index 0 CP. ) Additional discrete critical points in upper right are due to boundary artifacts.