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Learning Conjugate Direction Fields for Planar Quadrilateral Mesh Generation

Conference: AAAI 2026 arXiv: 2511.11865 Code: https://github.com/jiongtj/Learning-CDF Area: 3D Vision Keywords: Planar Quadrilateral Mesh, Conjugate Direction Field, Deep Learning, Architectural Design, Controllable Generation

TL;DR

This paper proposes a data-driven approach based on DGCNN to efficiently generate conjugate direction fields (CDFs), bypassing the high computational cost of traditional nonlinear optimization. The method supports user stroke-guided controllable CDF generation, achieves a 1–2 order-of-magnitude speedup, and is accompanied by a large-scale dataset of 50,000+ free-form surfaces.

Background & Motivation

Planar quadrilateral (PQ) meshes are critical in computer-aided design, particularly for the discretization of architectural surfaces. Their key advantages include: (1) planarity of faces significantly reduces manufacturing costs for physical materials such as glass; (2) lower vertex valence compared to triangle meshes reduces structural complexity; and (3) edge layouts are visually intuitive and aesthetically appealing.

PQ mesh generation typically follows a two-stage pipeline: first generating an initial quadrilateral mesh layout, then refining it via geometric optimization to make each face approximately planar. The quality of the initial layout depends on the conjugate direction field (CDF) defined on the surface — conjugacy of the CDF ensures that the initial mesh faces are approximately planar, which is critical for subsequent PQ mesh optimization.

Core Challenge: Unlike principal curvature direction fields (PDFs), which are uniquely determined by surface geometry (except at umbilic points), CDFs are non-unique and possess high degrees of freedom. Users must specify preferred directions via strokes, which serve as constraints in a nonlinear optimization to compute the CDF. However, this nonlinear optimization is: - Computationally expensive: cost grows sharply with mesh size (~17s for ~20k faces, ~40s for ~60k faces); - Iteratively demanding: designers frequently need to adjust strokes and recompute, resulting in poor interactivity; - An obstacle to exploration: real-time preview of PQ mesh layouts corresponding to different CDFs is infeasible.

Method

Overall Architecture

Input: Triangle mesh \(\mathcal{M} = \{\mathcal{V}, \mathcal{F}\}\) + user strokes \(\mathcal{S} = \{\mathbf{S}_i\}\)

Output: Per-triangle-face direction vector pairs \(\{(\mathbf{u}_j, \mathbf{v}_j)\}\) forming the CDF

Pipeline: Feature extraction → CDF prediction → Global parameterization → Quadrilateral mesh extraction → Vertex perturbation optimization

Key Designs

1. Feature Representation

A 9-dimensional feature vector is constructed for each vertex by concatenating three components:

  • Vertex position \(\mathbf{p}_i \in \mathbb{R}^3\): encodes mesh geometry
  • Vertex normal \(\mathbf{n}_i \in \mathbb{R}^3\): encodes local surface orientation
  • Stroke projection vector \(\mathbf{l}_i = \mathbf{p}_i^* - \mathbf{p}_i \in \mathbb{R}^3\): vector from the vertex to its nearest stroke point

The stroke projection vector serves as a global stroke representation — it encodes the spatial relationship between each vertex and the stroke curve, rather than processing the stroke in isolation. Experiments demonstrate that this representation substantially outperforms extracting stroke features via a Point Cloud Transformer (PCT) (δ: 8.31° vs. 20.98°).

2. Network Architecture

Feature Extraction Module: Based on DGCNN, using 4 EdgeConv layers to extract per-vertex features. Unlike the original DGCNN, this work concatenates each vertex's local features with global shape features, then applies a fully connected layer to obtain a 256-dimensional feature representation. DGCNN dynamically recomputes local neighborhoods at each layer, enabling adaptive learning of multi-scale geometric information.

Prediction Module: Two independent MLPs predict \(\{\mathbf{u}_j\}\) and \(\{\mathbf{v}_j\}\) separately. Per-face features are obtained by averaging vertex features. Each MLP consists of 3 layers (256→128→64→3), with BatchNorm+ReLU applied to the first two layers; the final layer outputs predictions directly, which are then normalized to unit length.

3. Loss Function Design

Five loss terms are carefully designed:

Direction Alignment Loss \(\mathcal{L}_d\): Measures alignment between predicted CDFs and ground truth. The 90°-rotated ground truth vectors are used to handle sign ambiguity, and the minimum over two possible correspondences is taken:

\[\mathcal{L}_d = \frac{1}{m}\sum_{j=1}^{m} \min(E_j, E_j')\]

where \(E_j = (\mathbf{u}_j \cdot \mathbf{u}_j^{*\perp})^2 + (\mathbf{v}_j \cdot \mathbf{v}_j^{*\perp})^2\)

Normal Consistency Loss \(\mathcal{L}_{dn}\): Ensures predicted directions are orthogonal to face normals:

\[\mathcal{L}_{dn} = \frac{1}{m}\sum_{j=1}^{m} (\mathbf{u}_j \cdot \mathbf{n}_j)^2 + (\mathbf{v}_j \cdot \mathbf{n}_j)^2\]

Direction Smoothness Loss \(\mathcal{L}_{ds}\): Enforces smooth CDF transitions across adjacent faces to reduce singularities:

\[\mathcal{L}_{ds} = \frac{1}{|\mathcal{N}|}\sum_{(j,k)\in\mathcal{N}} \min(E_{jk}, E_{jk}')\]

Parallel transport is used to handle differing face normals between adjacent faces.

Stroke Consistency Loss \(\mathcal{L}_{dc}\): Ensures CDF directions align with user strokes:

\[\mathcal{L}_{dc} = \frac{1}{|\mathcal{S}|}\sum_{\mathbf{S}_i \in \mathcal{S}} \frac{1}{|\mathcal{T}_i|}\sum_{k \in \mathcal{T}_i} D_k\]

Field Regularization Loss \(\mathcal{L}_{fr}\): Prevents degenerate zero-vector predictions:

\[\mathcal{L}_{fr} = \frac{1}{m}\sum_{j=1}^{m} (||\mathbf{u}_j||-1)^2 + (||\mathbf{v}_j||-1)^2\]

Total loss: \(\mathcal{L}_{total} = \mathcal{L}_d + \lambda_1\mathcal{L}_{dn} + \lambda_2\mathcal{L}_{ds} + \lambda_3\mathcal{L}_{dc} + \lambda_4\mathcal{L}_{fr}\)

All weights are set to 1.0.

Loss & Training

  • Dataset: 50,000 training + 2,500 validation + 300 test B-spline surfaces
  • Each surface has 2,601 sampled points and 5,000 faces
  • Position and orientation normalized via PCA
  • Adam optimizer, learning rate \(1.0 \times 10^{-4}\), trained for 200 epochs
  • Hardware: Intel i9-14900K + NVIDIA RTX 4090

Key Experimental Results

Main Results (Computational Efficiency Comparison)

CDF generation time comparison (vs. traditional optimization):

Model Faces Optimization Ours Speedup
Test Model 1 5,000 2.851s 0.200s 14.3×
Test Model 2 5,000 2.855s 0.194s 14.7×
Vase 23,642 17.326s 0.254s 68.2×
Dome 44,490 30.198s 0.417s 72.4×
Face 60,077 40.412s 0.571s 70.8×
Garden (arch.) 8,322 4.946s 0.206s 24.0×
Yas Island (arch.) 7,029 3.766s 0.204s 18.5×
Aqua Dome (arch.) 10,790 6.522s 0.217s 30.1×

Speedup increases with mesh size: ~15× at 5k faces, ~71× at 60k faces. The traditional method scales nearly linearly, while the learning-based method exhibits nearly flat scaling.

Ablation Study

Effect of ablating loss terms on the test set (averaged over 300 models):

Configuration Singularities δ (stroke consistency) θ (CDF proximity) Notes
Full model 4.91 8.31° 11.30° Complete model
w/o \(\mathcal{L}_{ds}\) 7.02 7.38° 10.55° Significant increase in singularities
w/o \(\mathcal{L}_{dc}\) 4.67 10.32° 11.48° Degraded stroke consistency
PCT stroke features 8.97 20.98° 19.06° Far inferior to proposed representation

Key Findings

  • Smoothness loss \(\mathcal{L}_{ds}\) is critical for reducing PQ mesh singularities (7.02→4.91)
  • Stroke consistency loss \(\mathcal{L}_{dc}\) reduces δ from 10.32° to 8.31°
  • The proposed stroke projection vector representation substantially outperforms PCT (60% reduction in δ, 41% reduction in θ), validating the importance of surface-context-aware encoding
  • PQ mesh planarity is already favorable at initialization (\(\eta_{\text{mean}} \approx 0.006\)) and further improves to ~0.002 after vertex perturbation optimization
  • The method generalizes to open-boundary surfaces, real architectural surfaces, and closed models with varied topology (e.g., Stanford Bunny)
  • Compared with VectorHeat and NeurCross: VectorHeat cannot guarantee conjugacy; NeurCross is restricted to PDFs — neither is suitable for controllable CDF generation

Highlights & Insights

  1. Precise problem formulation: The non-uniqueness of CDFs is both a source of flexibility and a computational bottleneck; using a learning approach to bypass nonlinear optimization is natural and effective.
  2. Clever stroke representation design: Projection vectors encode the spatial relationship between each vertex and the stroke, implicitly propagating stroke information across the entire surface.
  3. Elegant handling of direction ambiguity: The 90°-rotation and minimum-over-correspondences strategy gracefully resolves the inherent sign and correspondence ambiguity of direction fields.
  4. Large-scale dataset contribution: The synthetic dataset of 50,000+ samples is itself a practical contribution; training data construction simulates real design workflows by tracing streamlines from ground-truth CDFs to simulate user strokes.

Limitations & Future Work

  • CDFs cannot accurately align with sharp features, as training data does not include surfaces with such characteristics
  • No explicit control over the number or placement of singularities
  • Reliance on synthetic B-spline surface data may limit generalization
  • Unsupervised approaches for improved generalization remain unexplored
  • The influence of stroke coverage and density on results lacks systematic analysis
  • Traditional methods (Liu et al. 2011) compute CDFs via constrained nonlinear optimization; this work replaces that process entirely with learning
  • Sketch2PQ (Deng et al. 2022) predicts PQ meshes from 2D sketches but is limited in handling 3D surfaces
  • VectorHeatNet learns vector fields but does not guarantee conjugacy
  • Insight: Learning-as-optimization substitution may be equally applicable to other computational bottlenecks in architectural CAD, such as surface unfolding and structural optimization

Rating

  • Novelty: ⭐⭐⭐⭐ — First application of deep learning to CDF generation
  • Experimental Thoroughness: ⭐⭐⭐⭐ — Comprehensive efficiency comparisons, complete ablations, generalization tests on architectural and general 3D models
  • Writing Quality: ⭐⭐⭐⭐⭐ — Clear problem formulation and rigorous mathematical derivation
  • Value: ⭐⭐⭐⭐ — Direct applicability to architectural design CAD