Robustness-Based Simplification of 2D Steady and Unsteady Vector Fields

Abstract

Abstract-Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness which enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory and has minimal boundary restrictions. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets. We show local and complete hierarchical simplifications for steady as well as unsteady vector fields.

Robust Extraction and Simplification of 2D Symmetric Tensor Field Topology

“…It quantifies the stability of critical points with respect to the minimum amount of perturbation required to remove them. For the analysis and visualization of vector fields, robustness has been applied to 2D and 3D fields [SWCR14, SWCR15, SRW∗16]. A robustness‐based vector field simplification strategy has been introduced independent of the topological skeleton [SWCR14]; for 3D vector fields, robustness gives rise to the first simplification method using critical point cancellation [SRW∗16].…”

“…Comparison with Previous Work. The work in [SWCR15] establishes the theoretical and algorithmic framework for robustness‐based simplification of steady and unsteady vector fields. While the simplification frameworks of both vector and tensor fields are based on the well group theory, the one for tensor field requires additional, nontrivial algorithmic development (such as the study of tensor field perturbation and anisotropy vector field mapping, see Section 3.2).…”

“…Robustness for degenerate points quantifies their structural stability and can be used as an importance measure for the hierarchical simplification of tensor fields. Robustness for critical points has been applied successfully in the setting of vector field simplification [SWCR15]. Its theoretical foundation for the analytic tensor field simplification has been established and is summarized in Section 3.…”

“…The robustness quantifies the stability of degenerate points with respect to the minimum amount of perturbation in the tensor fields required to remove them. Although the robustness for vector field topology [CPS12, SRW∗16, SW14, SWCR15, WRS∗13] is a well‐founded framework based upon the well group theory [EMP10, EMP11], when transferring the theory of robustness for vector fields to tensor fields, the essential questions are: (i) how should perturbations of tensor fields be defined and how can they be interpreted as easily tractable properties of the fields; and (ii) how can we relate these properties to our features of interest, that is, the degenerate points.…”

A Survey of Topology‐based Methods in Visualization

“…Wang et al [WRS*13] provide a sophisticated framework to investigate robustness of critical points that encodes its stability under small perturbations; this allows effective separation of features from noise. Similarly, Skraba et al develop a simplification algorithm for time‐varying critical points based on degree theory [SWCR15].…”