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Simulator HC: Regression-based Online Simulation of Starting Problem-Solution Pairs for Homotopy Continuation in Geometric Vision
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Proposes Simulator HC, which regresses a rough initial pose from correspondences, regenerates a consistent starting problem via an online reprojection simulator, and tracks a single root with homotopy continuation for generalized PnP and generalized relative pose and scale.
Problem
Polynomial minimal problems in geometric vision (e.g., generalized camera resectioning and generalized relative pose and scale) become intractable for elimination templates, and homotopy continuation is expensive when tracking all solutions or requires good starting problem-solution pairs.
Approach
Instead of classifying starting problem-solution pairs from real data, regress a single approximate solution with a network trained on simulated data, then use an online simulator (reprojection of points under the predicted pose) to build a consistent starting problem whose solution is the prediction, and track only that one root via HC to the target problem.
Key results
- On generalized PnP, Pred Initial + Simulator HC reaches 100% success at 0.316 ms versus 1.254 ms for Gröbner Bases and 9.796 ms for CPU HC.
- On generalized relative pose and scale with 8 correspondences, Simulator HC reaches about 95% success and is about 17x faster than Gröbner Bases.
- Within RANSAC on LaMAR, Simulator HC attains higher success rates than Gröbner Bases on all three sequences with lower runtime (e.g., 84% vs 67% on HGE).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add predicted initialization and online simulation before single-root homotopy continuation | Generalized PnP · Noisy synthetic data | success rate: 32 -> 100 | Helped |
| Add online simulation and homotopy continuation after the predicted initialization | Generalized PnP · Noisy synthetic data | success rate: 54 -> 100 | Helped |
| Replace predicted initialization with random initialization for Simulator HC (ablation) | Generalized PnP · Noisy synthetic data | success rate: 100 -> 22 | Hurt |
| Add an eighth correspondence instead of using the minimal 7-point input (ablation) | Simulator HC for GRPS · Noise-free synthetic data | success rate: around 70% -> 95%; 96.3% | Helped |
| Use Simulator HC instead of Gröbner Bases in GRPS RANSAC | GRPS · RANSAC, 10%, 20%, and 30% outliers | Time (s): 2.33 | 0.85; 4.37 | 1.31; 5.10 | 1.35 | Helped |
| Use Simulator HC instead of Gröbner Bases in GRPS RANSAC | GRPS · RANSAC, 40% outliers | success rate: 98.9 | 91 | Hurt |
| Use Simulator HC instead of Gröbner Bases | GRPS · LaMAR CAB, HGE, and LIN | success rate: 60 -> 68; 67 -> 84; 72 -> 86 | Helped |
Surprising findings
- Tracking a single root from a regressed-then-simulated starting pair outperforms exhaustive parallel HC on GPU, which is about 5x slower than the single-track approach.
Limitations
- Minimal 7-correspondence setting yields only about 70% success, requiring an extra correspondence (8 points) to reach 95%.
- Regressor accuracy is modest; the method relies on the predicted pose being a good enough starting point.
- Evaluated only on two generalized camera problems and a single real dataset.
Experimental setup
Datasets: simulated correspondences LaMAR
Benchmarks: LaMAR
Metrics: success rate rotation error (deg) translation error scale error (%) runtime (ms) RANSAC iterations
Baselines: Gröbner Bases (UPnP) Levenberg-Marquardt CPU HC GPU HC Eigen solver Pred Initial + LM
Builds on: Homotopy continuation (Duff et al.) Learning to solve hard minimal problems (Hruby et al.) UPnP Monodromy homotopy
Closest papers by meaning
- GPU-Based Homotopy Continuation for Minimal Problems in Computer Vision (CVPR 2022 · similarity 0.91)
- TRPLP - Trifocal Relative Pose From Lines at Points (CVPR 2020 · similarity 0.87)
- Noisy One-point Homographies are Surprisingly Good (CVPR 2024 · similarity 0.84)
- Three-view Focal Length Recovery From Homographies (CVPR 2025 · similarity 0.84)
- Camera Resection from Known Line Pencils and a Radially Distorted Scanline (CVPR 2025 · similarity 0.83)
- GenHMR: Generative Human Mesh Recovery (AAAI 2025 · similarity 0.83)
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