Localized Evaluation for Constructing Discrete Vector Fields

Area of Science:

• Computational Topology
• Applied Mathematics
• Scientific Visualization

Background:

• Topological abstractions summarize vector field behavior but face numerical precision challenges.
• Discrete Morse theory offers an alternative using pairs of simplices, but general vector field computation is complex.
• Existing methods for general vector fields often involve computationally expensive optimization.

Purpose of the Study:

• To introduce a fast, novel approach for pairing simplices in 2D, time-independent triangulated vector fields.
• To develop an efficient algorithm that overcomes the limitations of current state-of-the-art methods.
• To couple the pairing method with feature extraction, simplification, and visualization.

Main Methods:

• Employs a local evaluation strategy inspired by discrete gradient field construction.
• Assigns a unique outward flow direction to each edge and vertex in the mesh.
• Develops a linear-time algorithm processing vertex neighborhoods sequentially.

Main Results:

• Achieves drastic improvements in running time compared to existing methods.
• Produces topological features comparable to current state-of-the-art algorithms.
• Demonstrates successful application of simplification to large, complex flow datasets.

Conclusions:

• The proposed linear-time algorithm provides an efficient and robust method for analyzing 2D vector fields.
• This approach simplifies the computation of topological features, making complex flow analysis more accessible.
• The method shows significant potential for applications in scientific visualization and data analysis.