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

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, thus eliminating the need for explicit curvature extraction. Extensive experiments demonstrate that NeurCross outperforms the state-of-the-art methods in terms of singular point placement, robustness against surface noise and surface undulations, and alignment with principal curvature directions and sharp feature curves.

(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.

Introduction

In this paper, (1) We propose NeurCross, the first self-supervised neural network for learning cross fields. (2) We implicitly enforce cross field alignment with principal curvature directions via an SDF-based shape operator, naturally addressing potential ambiguity. (3) We leverage an optimizable neural SDF as an underlying representation to coordinate requirements, dynamically adjusting to minor surface variations.

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 architecture. The circled "+" symbol denotes a data-combining operation.

Results

Quad meshes generated by NeurCross and four other methods on the table model in the ShapeNet dataset.

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.

Our method is evaluated and compared against various state-ofthe-art methods using data provided by IGM (left) and Dielen et al. [2021] (right).

Comparison of quad meshes generated by Power Fields, PolyVectors (left), IM (middle), and Quad Remesher (right).

Quad meshes generated by all the methods on two models from ShapeNet (the airplane model) and Thingi10K (the grayloc model). We also show the locations of singular points, where "# of Sings" denotes the number of singular points on each quad mesh.

The top row shows the cross field generated by our method and four other methods on a noisy input mesh. The bottom row shows the resulting quad meshes produced by each approach. Note that MIQ fails to produce a valid result for this input surface with noise.

Comparison of quad meshes generated by various methods for some challenging models, i.e. with high genus, thin shells, and non-orientable rings. Across all tests, the quad meshes generated by NeurCross consistently exhibit higher quality compared to those produced by other methods.

Quad meshes generated by our NeurCross at different resolutions. The low-resolution models contain fewer than 1,000 vertices, while the high-resolution models consist of over 5,000 vertices.