DOGE: Differentiable Bézier Graph Optimization for Road Network Extraction¶

Conference: ECCV2026
Paper: https://eccv.ecva.net/virtual/2026/poster/4405
PDF: https://media.eventhosts.cc/Conferences/ECCV2026/pdfs/5775.pdf
Area: Remote Sensing
Keywords: Road network extraction, Bézier graph, differentiable rendering, geometric regularization, topology optimization

TL;DR¶

DOGE reformulates road extraction from remote sensing imagery as mask-guided curve-graph optimization: DiffAlign adjusts continuous geometry while TopoAdapt edits discrete connectivity, achieving TOPO F1 scores of 84.58 on SpaceNet and 80.59 on City-Scale without curve-level ground truth and producing more compact Bézier road networks.

Background & Motivation¶

A road segmentation mask from a satellite image is not yet a map suitable for route planning. Such a map must identify connected roads, permissible junction connections, and the continuous course of bends. Methods such as Sat2Graph, RNGDet++, and SAMRoad primarily output polyline graphs. Adding vertices lets polylines approximate bends, but their representation cost grows with curvature detail, and tangents are not inherently continuous at vertices. Compact storage, local editing, and smooth geometric constraints are therefore difficult to achieve together.

Cubic Bézier curves are better suited to curved roads, but supervised control-point prediction introduces an annotation problem: the same road can be divided into different numbers of curves, with nonunique control points. Converting existing polyline annotations into curves embeds arbitrary partitioning rules in the training targets. Skeletonizing a road mask and fitting curves avoids that supervision, but small mask gaps and jagged boundaries can become fixed connectivity errors. Conversely, optimizing curves solely for pixel agreement through differentiable rendering can produce overlapping, self-intersecting curves that do not form a usable navigation graph.

DOGE separates two questions: what shape the current edges should take, and which edges and connections the graph should contain. Gradients suit the first question; explicit discrete structural edits are needed for the second. Either mechanism alone leaves errors that it cannot repair. Core idea: use the mask as a rendering target for a curve graph, constrain continuous fitting with geometric priors, and alternate connection, simplification, and edge-addition operations so that road geometry and topology evolve together rather than being finalized by a single skeleton-extraction pass.

Method¶

Overall Architecture¶

The input is a satellite image, from which SAM2, fine-tuned on the corresponding dataset's training split, produces a road segmentation mask. DOGE consumes this mask and optimizes each image separately rather than predicting the entire road graph in one forward pass. It initializes a graph using “Chord-based Bézier Parameterization,” aligns its rendering to the mask through “DiffAlign Geometric Alignment,” and introduces “TopoAdapt Topology Refinement” into the loop after warmup. The output is a Bézier graph with road widths; curves are sampled into polylines only for evaluation.

“Ground-truth-free” here specifically means that curve-level vector ground truth is unnecessary, not that the whole system is unsupervised. SAM2 still undergoes dataset-specific road-segmentation training. During graph optimization, geometric parameters receive pixel losses from the mask instead of being regressed toward manually specified control points.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Satellite image<br/>Fine-tuned SAM2 mask"] --> B["Chord-based Bézier Parameterization<br/>Initialize curve graph"]
    B --> C["DiffAlign Geometric Alignment"]
    C -->|Continue after warmup| D["TopoAdapt Topology Refinement"]
    D --> C
    C -->|Stable structure and loss| E["Bézier road network<br/>Sample polylines for evaluation"]

Key Designs¶

1. Chord-based Bézier Parameterization: optimize shape while retaining connectivity through shared endpoints

Graph nodes have movable two-dimensional positions and can represent intersections, road termini, or intermediate road points. Each edge is a cubic Bézier curve with its own learnable width. Curve endpoints are bound directly to graph nodes, so edges that share a node remain connected by construction; connectivity does not have to be inferred from whether rendered strokes appear to touch. Unlike lane-level Bézier graphs that bind endpoint tangents to node-shared directions, DOGE retains independent internal-control-point parameters for each edge to accommodate the more flexible junction structures of standard-definition road maps.

Rather than directly updating all coordinates of the four control points, DOGE locates the two internal points using a coordinate system of “projection along the endpoint chord plus perpendicular offset.” The following expression is organized from the textual definitions surrounding Eq. (2). The cached equation has damaged formatting, but the projection, normal offset, and endpoint bindings are explicitly described:

\[ \begin{aligned} \boldsymbol P_{k,0}&=\boldsymbol p_i,\qquad \boldsymbol P_{k,3}=\boldsymbol p_j,\\ \boldsymbol P_{k,r+1}&=(1-\alpha_{k,r})\boldsymbol p_i +\alpha_{k,r}\boldsymbol p_j+d_{k,r}\boldsymbol n_{ij}, \qquad r\in\{0,1\}. \end{aligned} \]

Here, \(\alpha_{k,r}\in[0,1]\) specifies the position along the chord, \(d_{k,r}\) is a signed perpendicular offset, and \(\boldsymbol n_{ij}\) is the unit chord normal. The optimizer adjusts how far a control point advances along the road and bends sideways rather than moving it arbitrarily. Combined with the offset and spacing regularizers below, this discourages ill-formed curves. These are structured parameters and soft constraints: they do not establish a mathematical guarantee against all inter-edge intersections, nor do they automatically make junctions higher-order continuous.

2. DiffAlign Geometric Alignment: let pixel errors update curves without rewarding invalid geometry

DiffAlign serializes width-bearing curved road segments as closed polygons and renders them with a differentiable rasterizer built on DiffVG. Gradients of pixel coverage flow first to control points, then through the parameterization to node positions, road widths, chord projections, and normal offsets. Roads can move as a whole while also changing curvature and width, unlike fitting to a fixed skeleton that already commits to the extracted geometry.

Pixel error alone is insufficient: several duplicate roads covering the same region may look acceptable in the final union image. DOGE's coverage loss measures pixel-wise squared error between the union of rendered edges and the target mask. Its overlap loss instead examines the sum of individual edge renderings, penalizes coverage beyond a single layer, and normalizes by the number of edges. These losses address different questions—whether road regions are covered, and whether duplicate or improperly intersecting roads are being used to obtain that coverage—and cannot replace one another.

Another prior is \(G^1\) continuity. At degree-2 nodes, it compares the ending tangent of the preceding edge with the starting tangent of the following edge to align nearly straight connections. The penalty activates only when the tangent angle falls below a threshold, avoiding the straightening of genuine turns. This is local tangent alignment, not a guarantee of curvature continuity throughout the road network. Finally, the offset loss discourages excessive lateral displacement relative to chord length, while the spacing loss pushes the two projections toward \(1/3\) and \(2/3\) along the chord, reducing control-point crowding and curve degeneration. The paper treats these terms as geometric priors independent of the target mask.

3. TopoAdapt Topology Refinement: edit the graph through connection, simplification, then expansion

Gradients can bring two endpoints closer, but cannot actually merge their nodes. Nor can they directly turn one road into two edges with a branch. TopoAdapt explicitly changes the node and edge sets. It first merges nearby nodes to remove redundant endpoints, then snaps a node near an edge onto that edge and splits it, creating a genuine T-junction. Merging before junction creation prevents repeated connections from being introduced while duplicate endpoints remain unresolved.

After consolidation, the algorithm removes redundant, nearly collinear degree-2 nodes and merges their incident edges. It then prunes excessively short or thin edges and isolated nodes. This does more than reduce node count: it keeps the optimizer from spending computation on fragments introduced by connectivity updates. Only then does it identify target-mask locations with sufficiently high confidence but insufficient rendered coverage, sample new edges there, and randomly offset their endpoints. The same edge-addition mechanism initializes the empty graph. Instead of distributing more curves everywhere, it concentrates new capacity on roads that the current graph misses.

A spatial grid accelerates these operations, which use fixed geometric thresholds. The main paper specifies a shared distance threshold of 4 m for node merging and T-junction creation. Importantly, DOGE avoids letting one-shot skeletonization and curve fitting determine the final graph; it does not eliminate every hand-designed rule. Its discrete operations still use thresholds, but subsequent geometric optimization can continue to refine the resulting structure, and the paper keeps topology hyperparameters consistent across both datasets.

Loss & Training¶

Segmentation fine-tuning and per-image curve optimization are distinct: the former supplies the observation target, while the latter uses Adam to adjust the current graph parameters. Using the five losses and weights listed in the main paper, the total objective is:

\[ \mathcal L_{\mathrm{total}} =\mathcal L_{\mathrm{cover}} +0.3\mathcal L_{\mathrm{overlap}} +0.012\mathcal L_{G1} +6\times10^{-3}\mathcal L_{\mathrm{offset}} +6\times10^{-3}\mathcal L_{\mathrm{spacing}}. \]

The main experiments use a 512×512 rendering resolution, at most 300 optimization iterations, and a single NVIDIA RTX 4090. Geometry is warmed up after initialization before recurring topology refinement begins. In the original algorithm, each post-warmup iteration first updates discrete connectivity and then optimizes geometry. Early stopping occurs when the loss change between successive iterations stays below a tolerance and the node-edge structure remains unchanged for the required patience count.

The main paper refers the learning rate, warmup duration, early-stopping tolerance, and remaining topology thresholds to supplementary material rather than listing all of them. The local full-text cache contains only the main paper and references. Accordingly, this note does not supply those missing values or infer the exact norm definitions of the offset and spacing losses from damaged equations.

Key Experimental Results¶

Main Results¶

City-Scale uses a 144/9/27 train/validation/test split, and SpaceNet uses 2042/127/382. Both are standardized to 1 m/pixel following prior work. Bézier edges are uniformly sampled into polylines before being passed to the public evaluation implementations. TOPO F1 combines the precision and recall of topological matching; APLS measures shortest-path-length agreement between predicted and ground-truth road graphs. All four metrics below are higher-is-better. Values come directly from Table 1 of the main paper and are not mixed with the separate identical-mask comparison discussed later.

Dataset	Method	TOPO F1 ↑	Precision ↑	Recall ↑	APLS ↑
SpaceNet	RNGDet++	82.81	91.34	75.24	67.73
SpaceNet	SAMRoad	80.52	93.03	70.97	71.64
SpaceNet	SAMRoad++	81.57	93.68	72.23	73.44
SpaceNet	DOGE	84.58	93.55	78.43	73.48
City-Scale	RNGDet++	78.44	85.65	72.58	67.76
City-Scale	SAMRoad	77.23	90.47	67.69	68.37
City-Scale	SAMRoad++	80.01	88.39	73.39	68.34
City-Scale	DOGE	80.59	84.42	77.40	70.24

On SpaceNet, TOPO F1 exceeds RNGDet++ by 1.77 percentage points, but APLS exceeds SAMRoad++ by only 0.04 percentage points; this is not a substantial lead on every metric. On City-Scale, recall improves by 4.01 percentage points over SAMRoad++, while precision declines by 3.97 percentage points. The F1 improvement therefore entails a clear precision trade-off.

Ablation Study¶

The following results come from Table 2 of the main paper, evaluated on SpaceNet. ED is the number of edges per kilometer of road, measured in edges/km, and describes representation compactness. Lower ED is meaningful only when topology quality remains adequate; missing roads should not be counted as beneficial compression. Both lower-bound baselines use the same SAM2 masks as the full model.

Config	TOPO F1 ↑	APLS ↑	ED ↓
Full DOGE	84.58	73.48	22.62
Without TopoAdapt	82.80	69.45	66.60
Without overlap loss	83.38	65.95	40.30
Without G1 loss	82.54	69.28	23.71
Without both curve regularizers	82.57	65.97	23.48
Without all geometric priors	74.58	51.18	44.17
Vanilla DiffRender	70.94	37.10	98.96
Standard Post-Processing	79.57	63.10	69.21

Key Findings¶

Removing TopoAdapt reduces APLS by 4.03 percentage points and raises ED from 22.62 to 66.60. Discrete edits improve both connectivity and compactness, not merely visual appearance.
Removing overlap loss reduces APLS by 7.53 percentage points, a larger decrease than removing TopoAdapt alone. Pixel coverage cannot replace structural constraints on road connectivity. Removing all geometric priors and using coverage-only optimization reinforce this conclusion, but ablation decreases cannot simply be added together.
In the additional comparison using identical SAM1 masks, DOGE/SAMRoad/SAMRoad++ achieve TOPO F1 scores of 83.20/80.08/79.68 and ED values of 21.33/47.12/47.64. This supports an independent benefit from vectorization, but these values should not be combined with rows in the main table that use different segmentation settings as though they belonged to one protocol.

Highlights & Insights¶

Differentiable rendering is a measurement bridge, not a complete solution. It converts visual agreement with roads into an optimization signal; geometric priors and discrete topology operations make low pixel error correspond to a usable road network.
Compactness depends on both curve expressiveness and structural simplification. Bézier edges represent bends with few parameters, while TopoAdapt removes redundant nodes. Merely switching to curves without topology updates still leaves high edge density in the ablation.
Identical-mask comparisons isolate the segmentation backbone's influence. They explain the source of gains more directly than the end-to-end leaderboard alone and avoid attributing all benefits of better SAM2 masks to the graph optimizer.

Limitations & Future Work¶

The authors identify severe occlusion, such as dense tree canopies, as a source of misleading masks and disconnected roads. Multilevel interchanges also lack height information needed to resolve actual connectivity. Two-dimensional rendering agreement alone cannot remove these ambiguities.
This is offline per-image optimization, not real-time one-pass inference. The main paper reports 302 min for DOGE-512 to process the SpaceNet test set on one RTX 4090. Fewer graph edges do not imply faster road extraction.
Evaluation primarily measures topology and paths after discretizing curves. The main paper refers further kinematic-smoothness and threshold-sensitivity analyses to supplementary material absent from the local cache. A curve representation alone does not establish comprehensive validation of downstream planning, curvature continuity, or latency.
This note distinguishes “no curve ground truth” from “no supervision” and “no heuristic rules.” The paper satisfies the first condition but still depends on segmentation fine-tuning, geometric priors, and topology thresholds. The authors' proposed occlusion priors, height cues, and efficiency improvements address these applicability boundaries.

vs SAMRoad/SAMRoad++: These methods provide foundation-segmentation-based road-graph extraction references. DOGE replaces mask-to-graph processing with constrained curve-graph optimization. Identical-mask experiments support the value of this backend difference, although the main leaderboard still includes different segmentation settings.
vs Bézier Everywhere All at Once: That work models lane-level Bézier graphs and constrains adjacent edges through node-shared directions. DOGE targets standard-definition road networks in remote sensing, using independent chord-based offsets per edge for greater geometric flexibility and avoiding supervision of curve control points.
vs DiffVG and skeleton fitting: DiffVG provides gradients from pixels to vector parameters, while skeletonization supplies discrete paths. DOGE's contribution is not to reinvent either tool, but to constrain gradient freedom with road priors and continually revise discrete structure rather than fixing errors after one extraction pass.

Rating¶

Novelty: 4/5. The combination of chord-based Bézier graphs, analytic geometric priors, and ordered topology edits yields a clear mask-driven road-optimization contribution, although the underlying tools have precedents.
Experimental Thoroughness: 4/5. Two main benchmarks, identical-mask controls, and detailed ablations are persuasive; some geometry and efficiency evidence still requires supplementary-material verification.
Writing Quality: 4/5. The roles of continuous geometry and discrete topology are clear, but readers must distinguish the scope of “ground-truth-free” and the small gains on some metrics.
Value: 4/5. Relevant to offline mapping that prioritizes compact, editable curve maps, without establishing suitability for real-time deployment.