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

TL;DR

NeurCross tackles quadrilateral mesh generation by jointly optimizing a neural signed distance function and a cross field, using the Hessian of the SDF to implicitly align cross-field directions with principal curvature directions while preserving cross-field smoothness. The framework comprises an SDF fitting module (SIREN) and a cross-field predictor (UNet), unified under a total loss that enforces surface fidelity, curvature alignment, and field coherence. By treating the neural SDF as a tunable proxy surface, NeurCross avoids brittle reliance on unstable principal directions and achieves robust performance across noisy and geometrically complex shapes. The results show improved singularity placement, robust cross-field behavior under perturbations, and faithful surface approximation, with practical quad extraction via libigl/libQEx. This self-supervised approach also offers potential as a data source for training future generative mesh models.

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...

Figures (29)

• Figure 1: Existing approaches typically rely on principal curvature directions as input. However, due to the inherent instability of these directions, current methods often prioritize the smoothness of the cross field at the cost of alignment with the principal curvature directions. To address this limitation, our approach avoids explicitly extracting principal curvature directions. Instead, we assess whether the cross field at each point can function as eigenvectors of the shape operator.
• Figure 2: (a) The input mesh and its ground-truth principal curvature directions. (b) Two-step optimization: by first precomputing an SDF that precisely fits the input shape, the subsequent optimization step still suffers from sensitivity to minor geometric variations, failing to yield the desired cross field. (c) Joint optimization: by treating the SDF as a proxy for the input shape, simultaneous optimization of the SDF and the cross field allows the SDF to approximate the input shape while remaining robust to minor geometric variations, resulting in the desired cross field. We visualize the fitting errors between the SDF surface and the original surface using a color-coded scheme.
• Figure 3: Given the input triangular surface in (a), starting with a randomly initialized cross field in (b), our NeurCross method produces a smooth cross field in (c) that is well aligned with the principal curvature directions of the input surface. We use the global-seamless parametrization from libigl to obtain a parametrization aligned with the computed cross field, and then use libQEx to extract the final quad mesh in (d).
• Figure 4: Our self-supervised network pipeline for representing cross fields in quad mesh generation. All layers in the network are implemented as multi-layer perceptrons (MLPs), with the SDF fitting module utilizing the SIREN SIREN architecture. The circled "+" symbol denotes a data-combining operation.
• Figure 5: The smoothness constraint of the cross field better controls the distribution of singularity points. From left to right: (a) an input triangular mesh; (b) a cross field computed with NeurCross; (c) the quad mesh extracted from the cross field.