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Revisiting Sampson Approximations for Geometric Estimation Problems
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Theoretical and empirical revisit of the Sampson approximation to geometric (reprojection) error, giving explicit tightness bounds for single quadratic/polynomial constraints and validating on two-view, three-view, vanishing point, and 2D/3D pose tasks.
Problem
Geometric estimation in computer vision requires measuring how well an observation agrees with a model; the exact geometric error is expensive to compute, and the Sampson approximation is widely used without theoretical guarantees on its tightness.
Approach
Derives explicit bounds relating the Sampson error to the true geometric error for one quadratic or polynomial constraint (and generalizes to multiple constraints by linearizing a locally full-rank subset of n-m constraints), showing the approximation is tight when the linearization point is close to the constraint set and the curvature is small.
Key results
- For two-view relative pose on IMC-PT, Sampson approximation has AUC of 0.999 up to 1 px versus 0.902 for symmetric epipolar error.
- For three-view error, the Sampson approximation of the pairwise epipolar constraints C3 reaches AUC 0.998 at sigma=1px, versus 0.764 for averaged pairwise Sampson errors.
- On 7Scenes 2D/3D refinement, the Sampson approximation matches full covariance-weighted optimization at lower thresholds while running in 2.4 ms versus 10.1 ms.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Sampson loss instead of algebraic, cosine, or symmetric epipolar loss for pose refinement | Two-view relative pose estimation · IMC-PT, MegaDepth-1500, ScanNet-1500 | AUC@10° pose error: 0.678 vs 0.673 / 0.732 vs 0.728 / 0.657 vs 0.654 | Helped |
| Use the joint Sampson approximation on three pairwise constraints (C3) instead of averaging pairwise errors | Three-view Sampson error · Three-view synthetic correspondences, σ = 1px / 5px / 10px | AUC@1px: 0.764 / 0.312 / 0.172 -> 0.998 / 0.961 / 0.882 | Helped |
| Use four constraints C4 instead of three pairwise constraints C3 (ablation) | Three-view Sampson error · Three-view synthetic correspondences, σ = 1px / 5px / 10px | AUC@1px: 0.998 / 0.961 / 0.882 -> 0.765 / 0.481 / 0.353 | Hurt |
| Use nine equations C9 instead of three pairwise constraints C3 (ablation) | Three-view Sampson error · Three-view synthetic correspondences, σ = 1px / 5px / 10px | AUC@1px: 0.998 / 0.961 / 0.882 -> 0.348 / 0.075 / 0.034 | Hurt |
| Use a mixed set of two quadratic and one cubic constraint instead of C3 (ablation) | Three-view Sampson error · Three-view synthetic correspondences, σ = 1px / 5px / 10px | AUC@1px: 0.998 / 0.961 / 0.882 -> 0.995 / 0.937 / 0.830 | Hurt |
| Include 3D-point uncertainty in pose refinement | 7Scenes pose refinement · 7Scenes, τ = 5px / 10px / 20px | AUC@5cm position error: 0.358 / 0.350 / 0.311 -> 0.367 / 0.379 / 0.378 | Helped |
| Replace full covariance-weighted 2D/3D optimization with its Sampson approximation | 7Scenes pose refinement · 7Scenes, τ = 5px / 10px / 20px | AUC@5cm position error: 0.367 / 0.379 / 0.378 -> 0.367 / 0.379 / 0.375 | No clear effect |
| Include matches with a larger inlier threshold (20px), increasing linearization error (ablation) | Sampson approximation for 2D/3D covariance-weighted reprojection · 7Scenes, τ = 20px | AUC@5cm position error: 0.378 -> 0.375 | Hurt |
Surprising findings
- Linearizing all four or nine three-view constraints (C4, C9) performs much worse than linearizing the three pairwise constraints C3, and the naive pseudo-Sampson variants perform poorly.
- Averaging pairwise two-view Sampson errors for three-view matches is significantly worse than jointly approximating the constraints.
Limitations
- Bounds for multiple constraints are only generalized in the supplementary material and only for quadratic constraints with linearly independent Jacobian rows.
- Bounds require assumptions such as the constraint being nearly linear (small rho/(2||J||)) or small |C(z)| relative to curvature.
- Approximation quality degrades for correspondences with large errors (e.g., 20 px thresholds).
Experimental setup
Datasets: IMC Phototourism 2021 MegaDepth-1500 ScanNet-1500 YUD+ NYU VP 7Scenes
Benchmarks: IMC Phototourism 2021 MegaDepth-1500 ScanNet-1500 7Scenes YUD+ NYU VP
Metrics: AUC approximation error vs geometric error pose error (rotation and translation) runtime (ms)
Baselines: Algebraic error Cosine error Symmetric epipolar error Epair (pairwise Sampson) Reprojection error with covariance DLT
Builds on: Sampson error Gauss-Newton optimal triangulation
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