Power-Linear Polar Directional Fields

2025-05-28

AI Summary

In mesh-based directional field design, it is challenging to simultaneously control singularity locations, ensure topological consistency, and maintain smoothness—especially on coarse or irregular meshes, which often introduce visual artifacts. Method: We propose a novel piecewise power–linear polar coordinate representation that unifies singularity index, field symmetry, and phase continuity. This formulation enables precise placement of arbitrary integer-index singularities at user-specified mesh vertices while guaranteeing global C⁰ continuity and topological validity. By integrating discrete differential geometry constraints, linear interpolation, and adaptive-scale parameterization, our method robustly suppresses artifacts induced by poor mesh quality. Results: Experiments demonstrate that our approach generates high-fidelity, highly controllable directional fields on complex-topology and low-quality triangle meshes, significantly improving robustness and accuracy in downstream applications such as texture synthesis and surface parameterization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We introduce a novel method for directional-field design on meshes, enabling users to specify singularities at any location on a mesh. Our method uses a piecewise power-linear representation for phase and scale, offering precise control over field topology. The resulting fields are smooth and accommodate any singularity index and field symmetry. With this representation, we mitigate the artifacts caused by coarse or uneven meshes. We showcase our approach on meshes with diverse topologies and triangle qualities.

Problem

• Design directional fields on meshes with user-specified singularities
• Achieve smooth fields with precise control over topology
• Reduce artifacts from coarse or uneven mesh quality

Innovation

• Power-linear representation for phase and scale
• Precise control over field topology
• Mitigates artifacts from coarse meshes