Learning Conjugate Direction Fields for Planar Quadrilateral Mesh Generation

TL;DR

The paper tackles the computational bottleneck of designing conjugate direction fields (CDF) for planar quadrilateral mesh generation by introducing a data-driven framework that learns to produce CDFs conditioned on user strokes. It employs a DGCNN-based encoder to fuse surface geometry with stroke guidance and two MLPs to predict the CDF components, optimized via a rich loss family enforcing alignment, tangentiality, smoothness, and stroke conformity. A large synthetic dataset of freeform surfaces with ground-truth CDFs and stroke mimics underpins training and evaluation, yielding substantial speedups over traditional optimization while maintaining high-quality planarity and stroke adherence. The work enables interactive, design-driven PQ mesh generation on complex 3D surfaces and offers a foundation for future enhancements in feature-aware CDF control and unsupervised generalization.

Abstract

Planar quadrilateral (PQ) mesh generation is a key process in computer-aided design, particularly for architectural applications where the goal is to discretize a freeform surface using planar quad faces. The conjugate direction field (CDF) defined on the freeform surface plays a significant role in generating a PQ mesh, as it largely determines the PQ mesh layout. Conventionally, a CDF is obtained by solving a complex non-linear optimization problem that incorporates user preferences, i.e., aligning the CDF with user-specified strokes on the surface. This often requires a large number of iterations that are computationally expensive, preventing the interactive CDF design process for a desirable PQ mesh. To address this challenge, we propose a data-driven approach based on neural networks for controlled CDF generation. Our approach can effectively learn and fuse features from the freeform surface and the user strokes, and efficiently generate quality CDF respecting user guidance. To enable training and testing, we also present a dataset composed of 50000+ freeform surfaces with ground-truth CDFs, as well as a set of metrics for quantitative evaluation. The effectiveness and efficiency of our work are demonstrated by extensive experiments using testing data, architectural surfaces, and general 3D shapes.

Figures (12)

• Figure 1: Visualization of direction fields and generated meshes on an ellipsoid.
• Figure 2: Our pipeline for inferring the conjugate direction field for PQ mesh generation. There are two core components in our learning model, including feature extraction and CDF prediction. In the first component, we take vertex positions, vertex normals, and the projected vectors to the user strokes as input, obtaining the latent feature for each vertex. In the second component, we use two MLPs to infer a CDF over the surface based on the latent features. In the figure, LBR means the combination of Linear, BatchNorm, and ReLU layers.
• Figure 3: PQ meshes generated using conjugate direction field inferred by our network on different types of surfaces as input. The blue lines on the surface are input strokes, which are streamlines traced from ground-truth CDF for comparison purposes. Left: B-spline surfaces from our test set; Middle: Open-boundary surfaces; Right: Real architectural surfaces. The input triangle meshes and strokes, as well as the output PQ meshes, are provided in the supplementary code and data.
• Figure 4: The examples where the different PQ mesh layouts are generated on the same surface with two different types of strokes as inputs.
• Figure 5: The time complexity illustration of the CDF generation between our method and the traditional optimization-based method diamanti2014designing.