TopoCurve: Geometry-Aware Topology Reasoning via Bézier Curves in Autonomous Driving¶
Conference: NeurIPS 2026
arXiv: 2610.09118
OpenReview: https://openreview.net/forum?id=peNZjVneDd
Area: Autonomous Driving
Keywords: autonomous driving, lane topology reasoning, Bézier curves, endpoint detection, deformable attention
TL;DR¶
Addressing the issues of jagged geometry, missing analytical tangents, and sparse topology supervision inherent in polyline representations, TopoCurve reparameterizes lanes as endpoint-fixed cubic Bézier curves, integrating Fourier-encoded geometric bias, curve-distributed deformable attention, and redundant supervision to establish a new state-of-the-art of 50.6 OLS on OpenLane-V2 without any post-processing.
Background & Motivation¶
In autonomous driving, online high-definition (HD) map construction and end-to-end trajectory planning require understanding not only the geometry of individual lane centerlines and traffic elements, but also their structural connectivity. Existing query-based frameworks predominantly model each 3D lane centerline as a polyline comprising 11 discrete 3D coordinates. However, independently regressing unstructured point coordinates discards geometric continuity, frequently producing jagged, self-intersecting, or physically implausible trajectories. Because lane connectivity is fundamentally determined by endpoint alignment and directional compatibility at merging or branching junctions, discrete polylines cannot provide stable analytical tangents at critical endpoints, while finite-difference estimations remain noisy and non-differentiable through bipartite matching.
Existing topology reasoning heads also suffer from severe geometric limitations. Methods relying purely on dot-product feature similarity completely ignore directional geometric alignment. Conversely, approaches that inject pairwise Euclidean distances directly into Graph Convolutional Network (GCN) adjacencies—such as TopoLogic—couple topology optimization with feature propagation, which can degrade primary lane detection quality. Furthermore, raw distance scalars struggle with neural tangent kernel spectral bias, failing to delineate sharp high-frequency decision boundaries between connected and unconnected lanes. In feature aggregation, standard deformable attention samples exclusively around a single centroid, leaving the critical endpoints of 30–50 m extended lanes unobserved. Compounding these weaknesses, one-to-one Hungarian matching against only ~10 ground-truth lanes yields valid topology supervision for under 0.2% of query pairs, resulting in extreme supervision sparsity.
Rather than patching endpoint misalignment via heuristic inference-time snapping like TopoPoint's PLGM, this paper addresses the root cause through representation reform. Core idea: reparameterize 3D lanes as endpoint-fixed cubic Bézier curves to exploit exact analytical endpoints and tangents via multi-scale Fourier-encoded geometric bias in the topology head, while anchoring curve-distributed deformable attention and one-to-many redundancy supervision along the parametric curve in a tightly coupled geometric cascade.
Method¶
Overall Architecture¶
TopoCurve processes multi-view camera inputs through a shared ResNet-50 backbone with FPN and constructs a unified Bird's-Eye-View (BEV) feature grid using a BEVFormer encoder. Front-view features are concurrently processed by a Deformable DETR head to predict 2D traffic element bounding boxes. The core point-lane decoder stacks 6 transformer layers to jointly update lane queries and endpoint queries: lane queries aggregate BEV features across the full lane using 4 parallel Bézier-Distributed Deformable Attention (BDDA) blocks, while endpoint queries use standard deformable attention. After concatenation and joint multi-head self-attention for global point-lane interaction, queries enter the Unified Scene Graph Network (USGN) for bidirectional GCN message passing. Lane queries regress 4 Bézier control points, topology heads predict connectivity using Fourier-enhanced geometric bias (FourierGeoBias), and pre-fusion intermediate queries provide curve-anchored redundancy supervision (CARS) during training.
%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
A["Multi-View Images<br/>ResNet-50 + FPN"] --> B["BEVFormer Encoder<br/>Construct BEV Feature Grid"]
B --> C["Endpoint-Fixed Bézier Reparameterization<br/>Predict 4 3D Control Points"]
C --> D["Bézier-Distributed Deformable Attention BDDA<br/>Full-Lane Geometry-Aligned Sampling"]
D --> E["Curve-Anchored Redundancy Supervision CARS<br/>4 Parallel Branches for O2M Supervision"]
E --> F["Unified Scene Graph Network USGN<br/>Point-Lane-Traffic Bidirectional GCN"]
F --> G["Fourier-Enhanced Geometry Injection FourierGeoBias<br/>Multi-Scale Tangent and Distance Bias"]
G --> H["Topology Prediction Output<br/>Lane-Lane and Lane-Traffic Adjacency"]Key Designs¶
1. Endpoint-Fixed Bézier Reparameterization: Compacting high-dimensional polylines into smooth analytical geometry
To resolve the structural inconsistencies and over-parameterization of regressing 11 unconstrained 3D coordinates (33 dimensions), TopoCurve parameterizes each lane as a cubic Bézier curve defined by 4 control points \(\mathbf{C}=[\mathbf{P}_0,\mathbf{P}_1,\mathbf{P}_2,\mathbf{P}_3]\in\mathbb{R}^{4\times 3}\), shrinking the regression target to 12 dimensions (a 63.6% reduction) while guaranteeing \(C^\infty\) smoothness. Unlike B-splines, cubic Bézier curves ensure that the first and last control points \(\mathbf{P}_0\) and \(\mathbf{P}_3\) mathematically coincide with the true lane endpoints. Because equidistant parametric sampling \(t_k\) clusters points in high-curvature regions rather than maintaining arc-length uniformity like OpenLane-V2 ground truth, the training pipeline pins \(\mathbf{P}_0\) and \(\mathbf{P}_3\) to the ground-truth endpoints and solves only the interior control points \(\mathbf{P}_1, \mathbf{P}_2\) via least-squares on the Bernstein basis \(\mathbf{A}_{\text{int}}\):
During both bipartite matching and L1 loss calculation, predicted control points \(\hat{\mathbf{C}}\) are projected back to 11 discrete 3D sample points \(\hat{\mathbf{L}}=\mathbf{A}\hat{\mathbf{C}}\) using the precomputed Bernstein basis matrix \(\mathbf{A}\), naturally allocating higher implicit supervision weights to high-curvature regions.
2. Fourier-Enhanced Directional Geometry Injection: Injecting analytical tangent alignment and high-frequency distance bias
To overcome the lack of directional awareness in dot-product similarity and the spectral bias of raw distance scalars, FourierGeoBias derives exact Euclidean endpoint distance \(d_{ij}=\|\mathbf{P}_{3,i}-\mathbf{P}_{0,j}\|_2\) and unit tangent cosine alignment \(\cos\theta_{ij}=\hat{\mathbf{t}}_{3,i}\cdot\hat{\mathbf{t}}_{0,j}\) analytically from the Bézier control points (where \(\hat{\mathbf{t}}_{3,i} = \frac{\mathbf{P}_{3,i}-\mathbf{P}_{2,i}}{\|\mathbf{P}_{3,i}-\mathbf{P}_{2,i}\|}\) and \(\hat{\mathbf{t}}_{0,j} = \frac{\mathbf{P}_{1,j}-\mathbf{P}_{0,j}}{\|\mathbf{P}_{1,j}-\mathbf{P}_{0,j}\|}\)). Both geometric scalars are mapped into high-frequency representations via multi-scale Fourier encoding \(\gamma(x)\) over \(L=8\) frequency bands:
Lane queries augmented with sinusoidal positional encoding are concatenated with the Fourier-encoded geometric features and mapped through a two-layer MLP with a zero-initialized final layer into a scalar bias \(b_{\text{geo}}[i,j]\). This bias is added directly to the lane-lane topology logit: \(\hat{G}^{ll}_{ij}=\mathbf{e}^s_i\cdot\mathbf{e}^s_j+b_{\text{geo}}[i,j]\). This decouples geometric priors from internal GCN feature propagation while exploiting the empirical fact that connected lane pairs exhibit average tangent cosine similarity exceeding 97%.
3. Bézier-Distributed Deformable Attention BDDA: Expanding receptive fields across the entire curve
Standard deformable attention learns offsets around a single centroid per query, leaving the boundary endpoints of 30–50 m lanes outside the effective receptive field. BDDA leverages the Bernstein projection of the current Bézier curve to supply \(K=11\) geometry-aligned reference locations spanning the full lane. Each attention head computes learned offsets and attention weights across all \(11 \times N\) sampling locations simultaneously, normalizing them with a joint Softmax to adaptively aggregate features along high-curvature or boundary segments. At each layer, predicted control points iteratively update the 11 reference locations for the subsequent layer, while layer 0 gracefully initializes with replicated centroids.
4. Curve-Anchored Redundancy Supervision CARS: Relieving sparse supervision via geometry-aware one-to-many training
Because typical driving scenes contain only ~10 ground-truth lanes, one-to-one Hungarian matching provides valid positive supervisory gradients to fewer than 0.2% of lane pairs. CARS instantiates \(M=4\) parallel BDDA blocks per decoder layer. Placed immediately before the self-attention suppression module, each block independently attends to BEV features around the 11 curve references, capturing diverse parametric and geometric features. During training, intermediate pre-fusion queries are extracted from each block; top-3 greedy matching assigns multiple positive candidates per ground-truth lane, and auxiliary lane-lane topology focal losses are computed in conjunction with FourierGeoBias. At test time, all auxiliary supervision branches are pruned, incurring zero inference overhead.
Loss & Training¶
The overall training objective combines multi-task detection and topology losses:
Detection objectives utilize focal loss for classification alongside L1 regression loss (evaluated in the 33-dimensional projected coordinate space for lanes) and GIoU loss for traffic elements. All topology predictions are supervised with focal loss. The auxiliary one-to-many loss \(\mathcal{L}_{\text{o2m}}\) averages auxiliary focal losses over all \(M \times L\) block-layer pairs. Loss coefficients match the baseline setup: \(\lambda_l=1.5\), \(\lambda_t=\lambda_p=1\), \(\lambda_{pl}=\lambda_{ll}=\lambda_{lt}=5\), and \(\lambda_{\text{o2m}}=2\). Training runs end-to-end for 24 epochs using AdamW with cosine annealing (initial learning rate \(3\times 10^{-4}\), weight decay 0.01) across two NVIDIA H200 GPUs with a batch size of 12.
Key Experimental Results¶
Main Results¶
Evaluated on the OpenLane-V2 benchmark across both Argoverse2 (subset_A) and nuScenes (subset_B) splits under standardized ResNet-50 backbones and 24-epoch schedules:
| Dataset | Method | DETl (Lane AP) | DETt (Element AP) | TOPll (Lane Topology) | TOPlt (Traffic Topology) | OLS (Overall Score) | DETp (Endpoint AP) |
|---|---|---|---|---|---|---|---|
| OpenLane-V2 (sub-A) | TopoNet | 28.6 | 48.6 | 10.9 | 23.8 | 39.8 | 43.8 |
| OpenLane-V2 (sub-A) | TopoMLP | 28.3 | 49.5 | 21.6 | 26.9 | 44.1 | 43.4 |
| OpenLane-V2 (sub-A) | TopoLogic | 29.9 | 47.2 | 23.9 | 25.4 | 44.1 | 45.2 |
| OpenLane-V2 (sub-A) | TopoFormer | 34.7 | 48.2 | 24.1 | 29.5 | 46.3 | – |
| OpenLane-V2 (sub-A) | TopoPoint (NeurIPS'25) | 31.4 | 55.3 | 28.7 | 30.0 | 48.8 | 52.6 |
| OpenLane-V2 (sub-A) | TopoCurve (Ours) | 33.9 | 55.6 | 30.7 | 33.0 | 50.6 | 56.8 |
| OpenLane-V2 (sub-B) | TopoPoint | 31.2 | 60.2 | 28.3 | 27.1 | 49.2 | 45.1 |
| OpenLane-V2 (sub-B) | TopoCurve (Ours) | 33.4 | 61.5 | 30.8 | 29.4 | 51.2 | 48.3 |
Ablation Study¶
Ablations on OpenLane-V2 subset_A evaluate incremental additions and leave-one-out drops from TopoPoint (Béz: Bézier representation; F: FourierGeoBias; BD: BDDA; CA: CARS):
| Config | DETl | TOPll | TOPlt | OLS | DETp | Note / Relative Drop |
|---|---|---|---|---|---|---|
| Full model | 33.9 | 30.7 | 33.0 | 50.6 | 56.8 | Full TopoCurve architecture |
| Add Béz only | 32.0 | 28.9 | 30.6 | 49.1 | 53.0 | Smooth parametric foundation |
| Add Béz + F | 32.2 | 30.1 | 31.3 | 49.6 | 53.8 | Directional alignment & Fourier bias |
| Add Béz + F + BD | 33.2 | 30.2 | 32.0 | 50.1 | 55.5 | Full-lane receptive field sampling |
| w/o Béz (Polyline) | 31.4 | 28.5 | 29.8 | 48.7 | 53.0 | Steepest degradation (-1.9 OLS), cascade collapses |
| w/o F (No Fourier bias) | 33.8 | 29.4 | 31.3 | 49.9 | 56.5 | Significant drop in topology (-1.3 TOPll) |
| w/o BD (Centroid attention) | 32.8 | 30.4 | 32.6 | 50.2 | 56.0 | Impaired endpoint & elongated lane feature learning |
| w/o CA (One-to-one only) | 33.2 | 30.2 | 32.0 | 50.1 | 55.5 | Sparse supervision limits query diversity |
Encoding ablations show that raw distance and tangent scalars achieve only 50.1 OLS, whereas 8-band Fourier encoding mitigates spectral bias to reach 50.6 OLS. Efficiency profiling on an NVIDIA A100 GPU shows TopoCurve operates at 13.1 FPS with 79.5M parameters and 1016.4 GFLOPs—delivering a 3.69% OLS gain over TopoPoint with only a 2.74% increase in computational cost.
Key Findings¶
- Bézier parameterization anchors the geometric cascade: Removing Bézier parameterization causes the steepest drop (-1.9 OLS), confirming that analytical continuity is essential for both downstream tangent extraction and curve-aligned attention.
- Intrinsic end-to-end alignment eliminates post-processing: While TopoPoint relies on Point-Lane Geometry Matching (PLGM) heuristic post-processing to snap endpoints (gaining +0.5 OLS), applying PLGM to TopoCurve actually degrades OLS from 50.6 to 50.5, proving that endpoint-fixed curves naturally resolve endpoint deviation inside the network.
- Analytical tangents resolve directional ambiguity: Fusing tangent alignment with endpoint distance provides strong discriminative power (connected centerlines exhibit >97% tangent cosine alignment), effectively preventing false connectivity predictions at branching and forking intersections.
Highlights & Insights¶
- Structured parametric curve replaces unstructured point regression: Using endpoint-fixed cubic Bézier curves compresses regression targets to 12 parameters while preserving exact endpoints, providing smooth trajectories ideally suited for downstream motion planners.
- Multi-scale Fourier encoding bypasses spectral bias: Encoding geometric distance and tangent alignment through 8 Fourier bands allows standard MLPs to capture high-frequency connectivity boundaries, decoupling geometric reasoning from feature learning.
- Pre-suppression parallel cross-attention enables zero-cost one-to-many supervision: Placing parallel BDDA branches before self-attention preserves candidate diversity for rich one-to-many supervision during training while incurring zero extra latency at inference.
Limitations & Future Work¶
- Single-frame temporal limitation: The current framework operates on single-frame multi-view images without temporal feature fusion, leaving room for improvement in distant occluded scenes.
- Fixed-degree limitation for intricate geometry: While cubic Bézier curves model standard roadway segments well, highly winding continuous S-curves may experience slight fitting errors, motivating future work on adaptive-degree curves or B-spline hybrids.
- Absence of Frenet frame integration: Geometric relationships are computed in Cartesian coordinates; incorporating vehicle kinematics and road-aligned Frenet coordinates could further optimize downstream planning integration.
Related Work & Insights¶
- vs TopoPoint (NeurIPS 2025): TopoPoint introduces endpoint queries and relies on heuristic PLGM post-processing snapping; TopoCurve enforces exact endpoint alignment directly via endpoint-fixed Bézier parameterization, beating TopoPoint by 4.2 points on DETp (56.8 vs. 52.6) purely end-to-end.
- vs TopoLogic (NeurIPS 2024): TopoLogic injects undirected Euclidean distances into GCN message passing, coupling geometry with feature learning; TopoCurve computes analytical directed tangent alignment and distance, injecting them as Fourier-encoded residual bias into the prediction head.
- vs RATopo (ACM MM 2025): RATopo applies one-to-many matching to centroid-based attention and dot-product heads; TopoCurve designs curve-anchored BDDA sampling alongside geometry-aware one-to-many supervision, ensuring complete architectural synergy.
Rating¶
- Novelty: ⭐⭐⭐⭐⭐ Seamlessly unifies endpoint-fixed Bézier geometry across representation, attention sampling, and supervision.
- Experimental Thoroughness: ⭐⭐⭐⭐⭐ Comprehensive benchmarks across OpenLane-V2 sub-A and sub-B with exhaustive component and encoding ablations.
- Writing Quality: ⭐⭐⭐⭐⭐ Clear mathematical formulations, rigorous motivation, and strong narrative cohesion.
- Value: ⭐⭐⭐⭐⭐ Eliminates brittle heuristics and discrete polyline artifacts, setting a robust geometric foundation for autonomous driving topology reasoning.