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An N-Point Linear Solver for Line and Motion Estimation with Event Cameras
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Proposes a linear N-point solver for event-camera line and partial linear velocity estimation using an angle-axis line parametrization, plus a geometric velocity averaging scheme across lines, integrated into GC-RANSAC.
Problem
Line-based motion estimation with event cameras requires solving a polynomial system (minimal, root-finding, unstable, slow) for line parameters and partial linear velocity, and fusing multiple lines to recover full velocity.
Approach
Using a singularity-free, three-DoF angle-axis (rotation-matrix) line parametrization makes the event incidence constraint linear in the unknowns, so N≥5 events can be solved by a single SVD; velocities from multiple lines are fused via a geometrically motivated homogeneous linear system.
Key results
- Linear solver runs in 3.25 µs average for five events versus 2046 µs for the Gröbner solver, over 600 times faster.
- In noise-free simulation the linear solver shows 0.00% error rate at ≥0.1° and ≥1.0° while the Gröbner solver has 1.00% and 0.28%.
- On real VECtor sequences, linear with non-linear optimization reaches mean errors of 16.5°, 22.1°, 19.9° versus 33.5°, 26.4°, 31.5° for the Gröbner solver.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the Gröbner solver with the linear solver | Line and motion estimation from event cameras · Synthetic solver comparison | average runtime (µs): 2046 -> 3.25 µs | Helped |
| Replace the Gröbner solver with the linear solver | Line and motion estimation from event cameras · Synthetic noise-free solver comparison | error rate ≥0.1° (%): 1.00 -> 0.00 | Helped |
| Use the linear solver instead of the Gröbner solver | Line and motion estimation from event cameras · Synthetic pixel noise (0.5 px), 5 events | direction error (mean, median; °): 7.80/1.67 -> 5.53/1.24 | Helped |
| Use the linear solver instead of the Gröbner solver | Line and motion estimation from event cameras · Synthetic timestamp jitter (0.5 ms), 5 events | direction error (mean, median; °): 3.61/0.83 -> 2.87/0.73 | Helped |
| Use the linear solver instead of the Gröbner solver | Line and motion estimation from event cameras · Synthetic gyroscope noise (5.0 °/s), 5 events | direction error (mean, median; °): 7.48/3.09 -> 6.53/2.47 | Helped |
| Increase the event count from 5 to 10 (ablation) | Linear solver · Synthetic pixel noise (0.5 px) | direction error (mean, median; °): 5.53/1.24 -> 0.46/0.15 | Helped |
| Increase the event count from 5 to 10 (ablation) | Linear solver · Synthetic timestamp jitter (0.5 ms) | direction error (mean, median; °): 2.87/0.73 -> 0.17/0.12 | Helped |
| Increase the event count from 5 to 10 (ablation) | Linear solver · Synthetic gyroscope noise (5.0 °/s) | direction error (mean, median; °): 6.53/2.47 -> 1.50/1.17 | Helped |
| Use the linear-only method instead of CELC+opt | Event-camera motion estimation · VECtor mountain-normal | direction error (mean, median; °): 27.7/29.3 -> 25.2/21.4 | Helped |
| Add non-minimal solver refinement (ablation) | Linear solver · VECtor mountain-normal | direction error (mean, median; °): 25.2/21.4 -> 17.0/17.2 | Helped |
| Add nonlinear optimization refinement (ablation) | Linear solver · VECtor desk-normal | direction error (mean, median; °): 22.7/23.4 -> 22.1/20.7 | Mixed |
| Add nonlinear optimization refinement (ablation) | Linear solver · VECtor sofa-normal | direction error (mean, median; °): 21.9/17.6 -> 19.9/15.0 | Helped |
Surprising findings
- The Gröbner polynomial solver fails to converge within 0.1° about 1% of the time in noise-free simulation due to elimination template instability.
Limitations
- Assumes first-order (constant linear velocity) dynamics within a short time window, with rotation from IMU gyroscope integration.
- Gyroscope noise is not averaged out by the solver, leaving residual error from angular velocity noise.
- Velocity fusion ignores uncertainty of partial velocity observations; parallel lines make the averaging degenerate.
Experimental setup
Datasets: VECtor Benchmark synthetic event simulation
Benchmarks: VECtor Benchmark (mountain-normal, desk-normal, sofa-normal) synthetic event simulation
Metrics: direction error (deg) runtime (µs) error rate at 0.1° and 1.0° thresholds
Baselines: Gröbner solver (Gao et al.) CELC+opt
Builds on: Gao et al. 2023 Grobner basis eventail solver GC-RANSAC NAPSAC CELC
Closest papers by meaning
- Full-DoF Egomotion Estimation for Event Cameras Using Geometric Solvers (CVPR 2025 · similarity 0.88)
- A Unifying Contrast Maximization Framework for Event Cameras, With Applications to Motion, Depth, and Optical Flow Estimation (CVPR 2018 · similarity 0.86)
- Single Image Optical Flow Estimation With an Event Camera (CVPR 2020 · similarity 0.85)
- Camera Resection from Known Line Pencils and a Radially Distorted Scanline (CVPR 2025 · similarity 0.83)
- TimeTracker: Event-based Continuous Point Tracking for Video Frame Interpolation with Non-linear Motion (CVPR 2025 · similarity 0.83)
- Event-3DGS: Event-based 3D Reconstruction Using 3D Gaussian Splatting (NeurIPS 2024 · similarity 0.83)
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