NeurCross: A Neural Approach to Computing Cross Fields for Quad Mesh Generation

2024-05-22

AI Summary

Quadrilateral mesh generation faces key challenges including noise sensitivity in principal curvature direction estimation, ambiguous definitions of spherical/planar regions, and unstable alignment between cross fields and underlying geometry. Method: This paper introduces a novel neural-symbolic paradigm that jointly optimizes a signed distance function (SDF) and a cross field. Geometry reconstruction and directional field design are unified into a single end-to-end differentiable optimization problem. The SDF’s Hessian is leveraged to implicitly compute robust curvature estimates, which regularize the cross field; additionally, curvature-aware smoothing constraints derived from the SDF are incorporated. Results: Experiments demonstrate substantial improvements in cross-field stability and curvature alignment accuracy—particularly under geometric noise, on near-planar regions, and where Gaussian curvature vanishes or remains constant. The resulting quadrilateral meshes exhibit superior feature preservation, singularity control, and topological robustness compared to state-of-the-art methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Localization, Mapping, and Navigation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Multilingual and cross-lingual Web search
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Quadrilateral mesh generation plays a crucial role in numerical simulations within Computer-Aided Design and Engineering (CAD/E). Producing high-quality quadrangulation typically requires satisfying four key criteria. First, the quadrilateral mesh should closely align with principal curvature directions. Second, singular points should be strategically placed and effectively minimized. Third, the mesh should accurately conform to sharp feature edges. Lastly, quadrangulation results should exhibit robustness against noise and minor geometric variations. Existing methods generally involve first computing a regular cross field to represent quad element orientations across the surface, followed by extracting a quadrilateral mesh aligned closely with this cross field. A primary challenge with this approach is balancing the smoothness of the cross field with its alignment to pre-computed principal curvature directions, which are sensitive to small surface perturbations and often ill-defined in spherical or planar regions. To tackle this challenge, we propose NeurCross, a novel framework that simultaneously optimizes a cross field and a neural signed distance function (SDF), whose zero-level set serves as a proxy of the input shape. Our joint optimization is guided by three factors: faithful approximation of the optimized SDF surface to the input surface, alignment between the cross field and the principal curvature field derived from the SDF surface, and smoothness of the cross field. Acting as an intermediary, the neural SDF contributes in two essential ways. First, it provides an alternative, optimizable base surface exhibiting more regular principal curvature directions for guiding the cross field. Second, we leverage the Hessian matrix of the neural SDF to implicitly enforce cross field alignment with principal curvature directions...

Problem

• Balancing cross field smoothness and curvature alignment
• Improving quad mesh robustness to noise
• Optimizing SDF and cross field jointly

Innovation

• Joint optimization of cross field and neural SDF
• Neural SDF provides regular principal curvature directions
• Hessian matrix enforces cross field alignment