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Three-view Focal Length Recovery From Homographies
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Proposes closed-form minimal solvers (Sturm sequences and hidden-variable polynomial eigenvalue methods) that recover one or two focal lengths from three-view homographies of a planar scene, plus a RANSAC pipeline and a new planar-scene dataset.
Problem
Recovering camera focal lengths from three views observing a single plane, where two-view geometry is insufficient and existing three-view solvers are very slow or high-degree.
Approach
Enforces consistency of the plane normal vector across two homographies, and uses elimination ideals to derive explicit degree-6 constraints on the entries of the symmetric matrices Q_j, yielding low-degree univariate or two-variable polynomial systems that are solved efficiently.
Key results
- Hfff + P3P reaches median focal error 0.0439 and mAA_f(0.1) 51.49 on real planar data, versus 0.0480 and 48.73 for the Heikkilä-based variant.
- The Hfff solver runs in about 17.3 µs versus 1404 µs for Heikkilä's solver, roughly 20x faster in the RANSAC setting (1102 ms vs 49 ms per run).
- Two-view baselines fEf and Ef perform much worse on fully planar scenes (median ξf ≈ 0.46–0.68).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use three-view homography solver Hfff + P3P instead of two-view baselines on planar scenes | Case I focal-length estimation · Planar scenes, Case I | median focal-length relative error ξf: 0.4793 -> 0.0439 | Helped |
| Use three-view homography solver Hfff + P3P instead of Hfff + P3P [23] | Case I focal-length estimation · Planar scenes, Case I | median focal-length relative error ξf: 0.0480 -> 0.0439 | Helped |
| Use three-view homography solver Hfff + P3P instead of Hfff + P3P [23] | Case I focal-length estimation · Planar scenes, Case I | runtime (ms): 1102.48 -> 49.31 | Helped |
| Use three-view homography solver Hff + P3P instead of two-view baselines | Case II focal-length estimation · Planar scenes, Case II | median focal-length relative error ξf: 0.4623 -> 0.0611 | Helped |
| Use three-view homography solver Hff + P3P instead of triplet baselines | Case II focal-length estimation · Planar scenes, Case II | median focal-length relative error ξf: 0.0691 -> 0.0611 | Helped |
| Use three-view homography solver Hff + P3P instead of triplet baselines | Case II focal-length estimation · Planar scenes, Case II | runtime (ms): 31.81 -> 35.97 | Mixed |
| Use three-view homography solver Hfff + P3P on scenes with a large planar region | Case I focal-length estimation · Synthetic scenes with varying planar-point proportion | focal-length error ξf | Helped |
| Use three-view homography solver Hff + P3P on planar scenes | Case II focal-length estimation · Synthetic planar scenes | focal-length error ξf | Helped |
| Use pairwise two-view solvers without off-plane points on fully planar scenes | fEf, fEf + PP, and Ef · Synthetic fully planar scenes | focal-length error ξf | Hurt |
Surprising findings
- Two-view solvers (fEf, fEf+PP, Ef) fail on fully planar scenes, while three-view homography solvers succeed.
- Heikkilä's three-view solver, though theoretically complete, yields worse accuracy than the proposed solver, possibly due to numerical stability.
Limitations
- Pure translation is degenerate for the equal-focal solver Hfff and Hfρρ solvers.
- For Cases III and IV, focal lengths cannot be distinguished using planar points alone; off-plane points are needed for scoring.
- Robust voting is unsuitable because multiple solver solutions can be geometrically feasible.
Experimental setup
Datasets: synthetic planar scenes newly collected planar-scene dataset (1870 images, 14 cameras, 6 scenes)
Benchmarks: synthetic planar scenes custom real planar-scene dataset
Metrics: median relative focal length error mean average accuracy mAA_f(t) runtime (ms) numerical stability (log10 focal error)
Baselines: Heikkilä [23] three-view homography solver fEf 6-point solver [27] fEf + PP [42] Ef 5-point/single-side solver [5] fEf + P3P [43] Ef + P3P
Builds on: Heikkilä three-view focal length solver PoseLib LO-RANSAC six-point partially calibrated relative pose solver
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