Source-linked AI summary
End-to-end Lane Detection through Differentiable Least-Squares Fitting
Wouter Van Gansbeke, Bert De Brabandere, Davy Neven, Marc Proesmans, Luc Van Gool
TL;DR
The paper addresses the mismatch between segmentation-focused training and lane-parameter estimation in conventional two-step lane detection. It uses a deep network to generate lane-specific weight maps and a differentiable weighted least-squares layer to fit curves end to end. The method outperforms a two-step procedure, while operating at about 70 fps, but fixed lane-map ordering limits handling of lane changes.
Problem
Two-step lane detection optimizes segmentation rather than the final lane curvature parameters, leaving model fitting separate from feature learning.
Method
A deep network generates nonnegative, lane-specific pixel weight maps whose coordinates are passed to a differentiable weighted least-squares fitting module and geometric loss.
Results
The end-to-end method outperforms a two-step procedure despite weaker supervision and operates at 70 fps.
Takeaways & Limitations
Joint optimization adapts features to lane-parameter estimation and produces inspectable segmentation-like weight maps rather than a wholly opaque pipeline.
Takeaways & Limitations
Fixed, predefined ordering of output weight maps makes lane changes hard to handle, and instance-segmentation methods achieve slightly higher accuracy in table 2.
Abstract
from arXiv · showhide
Lane detection is typically tackled with a two-step pipeline in which a segmentation mask of the lane markings is predicted first, and a lane line model (like a parabola or spline) is fitted to the post-processed mask next. The problem with such a two-step approach is that the parameters of the network are not optimized for the true task of interest (estimating the lane curvature parameters) but for a proxy task (segmenting the lane markings), resulting in sub-optimal performance. In this work, we propose a method to train a lane detector in an end-to-end manner, directly regressing the lane parameters. The architecture consists of two components: a deep network that predicts a segmentation-like weight map for each lane line, and a differentiable least-squares fitting module that returns for each map the parameters of the best-fitting curve in the weighted least-squares sense. These parameters can subsequently be supervised with a loss function of choice. Our method relies on the observation that it is possible to backpropagate through a least-squares fitting procedure. This leads to an end-to-end method where the features are optimized for the true task of interest: the network implicitly learns to generate features that prevent instabilities during the model fitting step, as opposed to two-step pipelines that need to handle outliers with heuristics. Additionally, the system is not just a black box but offers a degree of interpretability because the intermediately generated segmentation-like weight maps can be inspected and visualized. Code and a video is available at github.com/wvangansbeke/LaneDetection_End2End.
1. Introduction and Related Work
The paper integrates classical geometric fitting into deep networks, replacing multi-stage lane detection with an end-to-end differentiable architecture. It positions this design within broader efforts to combine learned features with geometric computer-vision knowledge and differentiable optimization.
- Context: The work applies a broader deep-learning trend of embedding prior computer-vision knowledge into architectures and losses to lane detection.This integration aims to combine engineered components’ parameter efficiency with learned features, while requiring classical methods to be reformulated differentiably.
- Contribution: The proposed lane detector replaces separate feature extraction and model fitting with a deep network followed by an in-network differentiable least-squares module.The resulting lane model parameters are supervised with a geometric loss, allowing the feature extractor and fitting procedure to be trained jointly.
- Contribution: Joint training lets features adapt to the final lane-parameter task and helps prevent outliers during model fitting, unlike independently optimized multi-stage pipelines.The system also exposes segmentation-like intermediate weight maps that can be inspected and visualized, providing a degree of interpretability.
- Related work: Most existing lane detectors use multi-stage pipelines with separate feature extraction and model fitting, including methods that use CNNs for feature extraction.The authors aim to show that their uncomplicated end-to-end parameter estimation outperforms a multi-step procedure rather than competing with highly optimized systems.
- Related work: The fitting module is a weighted least-squares optimization layer, connecting the method to differentiable in-network optimization and exact quadratic-program solving.Its contribution is demonstrating the effectiveness of such an optimization step on a real-world computer-vision task.
2. Method
The method integrates a deep network, differentiable weighted least-squares fitting, and a geometric loss to estimate lane-curve parameters end to end. It generates weighted pixel coordinates for each lane line, fits a chosen curve, and backpropagates parameter or geometric losses through the fitting procedure.
- 2.1. Generating Weighted Pixel Coordinates with a Deep Network: The network predicts nonnegative weight maps for lane lines, converting fixed pixel coordinates into weighted coordinate triplets for curve fitting.The output is squared to restrict weights to nonnegative values, and one map is generated for each lane line.
- 2.2. Weighted Least-Squares Fitting Module: The fitting module interprets weighted pixel coordinates as 2D observations and outputs the parameters of a best-fitting curve, such as a parabola or spline.The weighted formulation uses a diagonal matrix of observation weights and can be solved in standard least-squares form.
- 2.2. Weighted Least-Squares Fitting Module: Backpropagation through differentiable matrix operations lets the network optimize curve parameters directly instead of a proxy lane-segmentation objective.Derivatives of the fitted parameters with respect to the weight matrix and network parameters can be computed through the fitting procedure.
- 2.3. Geometric Loss Function: The fitted parameters can be supervised with mean squared error, but parameter errors have unequal effects on curve shape.This motivates a geometric loss that minimizes the squared area between predicted and ground-truth curves up to a point t.
- 2.2. Weighted Least-Squares Fitting Module: A known homography can transform the weighted coordinates into an orthographic reference frame, where lane lines are better approximated as parabolas.Only the coordinate list is transformed; the input image itself is not.
3. Experiments
The experiments test differentiable least-squares fitting in toy and lane-detection settings, comparing end-to-end training with conventional segmentation-based pipelines. End-to-end training achieves lower error and improves robustness and efficiency, while fixed lane-map ordering limits lane-change handling.
- 3.1. Toy Experiment: The toy experiment shows that backpropagation through weighted least-squares fitting can move coordinates, weights, or both toward a target curve.For lane detection, coordinates remain fixed and gradients update only the network-generated weights.
- 3.2. Ego-lane Detection: The end-to-end method converges to lower error on both training and validation sets than cross-entropy training, although convergence is slower.The comparison uses normalized area between ground-truth and predicted curves, with geometric loss as the optimized metric.
- 3.2. Ego-lane Detection: The learned weight maps focus on visible lane markings, handle marking variation, and correctly ignore faded markings.This qualitative behavior emerges despite the weaker supervision signal of direct curve-parameter training.
- 3.3. Multi-lane Detection: 0.7% lower accuracy error is achieved by end-to-end training than by the conventional segmentation-and-fitting baseline.The baseline trains with cross-entropy and fits lines as post-processing; the end-to-end method optimizes final coordinates directly.
- 3.3. Multi-lane Detection: 71 fps is achieved on an NVIDIA 1080Ti because line coordinates require no post-processing.The method also does not require expensive segmentation ground truth.
- 3.3. Multi-lane Detection: Fixed, predefined ordering of output weight maps makes lane changes difficult and leaves instance-segmentation methods with slightly higher accuracy in Table 2.Each output map corresponds to a weighted least-squares problem for a specific lane line.
4. Conclusion
The method estimates lane curvature by backpropagating through an in-network weighted least-squares fit, improving lane detection over a two-step procedure at 70 fps.
- The end-to-end method outperforms a two-step procedure despite using a weaker supervision signal.The experiment evaluates lane detection using an area-based geometric loss.
- Weighted least-squares fitting is performed in-network using image-conditioned weights generated by a deep network.
- The approach improves performance at 70 fps and may extend to other computer vision tasks with alternative differentiable fitting modules.Active contour models are given as one possible future application, while replacing least squares with differentiable gradient descent is left for future work.