Research map / Learning theory and algorithms / Matrix and tensor methods
Robust Self-calibration of Focal Lengths from the Fundamental Matrix
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Proposes an iterative Lagrange-multiplier method that jointly refines two focal lengths and principal points from a fundamental matrix (with priors), plus a cheap imaginary-focal-length rejection check inside RANSAC.
Problem
Self-calibrating two cameras' focal lengths from a fundamental matrix: the Bougnoux closed form is unstable, yields imaginary focal lengths, and is sensitive to noise in F and to principal point errors, especially near degenerate coplanar-principal-axis configurations.
Approach
Formulate focal length and principal point estimation as a constrained optimization with priors and Kruppa-equation constraints, solved iteratively (triangulation-style) where each step solves a degree-four two-variable polynomial system via Gröbner basis, avoiding Bougnoux initialization; additionally reject RANSAC models yielding imaginary squared focal lengths using a modified Bougnoux formula.
Key results
- On Phototourism, the proposed method reaches median focal length error 0.149 and mAAf 23.95 (vs Bougnoux 0.215 and 20.65), and 40.37 mAAp at 10 degrees pose accuracy.
- On Aachen Day-Night v1.1, the method achieves mAAf 12.46 and mAAp 27.55 vs Bougnoux mAAf 11.37 and mAAp 26.73, with RFC improving to 12.73 and 29.30.
- Mean runtime on Phototourism is 3.59 ms for the C++ Eigen implementation vs 659.61 ms for Hartley and Silpa-Anan (PyTorch).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the proposed iterative focal-length estimation method instead of the Bougnoux formula | two-camera self-calibration · Phototourism | mAAf (0.1): 20.65 -> 23.95 | Helped |
| use the proposed iterative focal-length estimation method instead of the Bougnoux formula | two-camera self-calibration · Aachen Day-Night v1.1 | mAAf (0.1): 11.37 -> 12.46 | Helped |
| add real focal length checking within RANSAC (ablation) | proposed method · Phototourism | mAAf (0.1): 23.95 -> 24.23 | Helped |
| add real focal length checking within RANSAC (ablation) | proposed method · Aachen Day-Night v1.1 | mAAp (10°): 27.55 -> 29.30 | Helped |
| add real focal length checking within RANSAC (ablation) | proposed method · Phototourism | mean runtime (ms): 3.59 -> 3.44 | Helped |
| assume equal focal lengths and use the proposed iterative method | two-camera self-calibration · ETH3D Multiview | mAAp (10°): 16.42 -> 16.70; 20.64 -> 16.70 | Mixed |
| replace the proposed method with Hartley and Silpa-Anan's iterative method | two-camera self-calibration · Phototourism | mAAf (0.1): 23.95 -> 12.73 | Hurt |
Surprising findings
- Fundamental matrices producing imaginary focal lengths under the Bougnoux formula rarely end up as the final RANSAC model, so rejecting them saves time and often improves accuracy.
Limitations
- Relies on accurate-ish priors and weight tuning (weights chosen on the Brandenburg Gate validation scene).
- Still operates on fundamental matrices from RANSAC; the Gröbner solver step is complex and the Matlab implementation is unoptimized.
- Ground-truth intrinsics from COLMAP may carry bias.
Experimental setup
Datasets: Phototourism Aachen Day-Night v1.1 ETH3D High Resolution Multiview
Benchmarks: Phototourism Aachen Day-Night v1.1 ETH3D High Resolution Multiview
Metrics: median pose error mAAp median focal length error mAAf mean runtime
Baselines: Bougnoux formula Hartley and Silpa-Anan Fetzer et al. Sturm Minimal 6-point Prior
Builds on: Bougnoux formula Hartley and Silpa-Anan iterative focal estimation MAGSAC++ LoFTR
Closest papers by meaning
- Linear Fundamental Matrix Estimation from 7 or 5 Points (CVPR 2026 · similarity 0.84)
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- Camera Resection from Known Line Pencils and a Radially Distorted Scanline (CVPR 2025 · similarity 0.82)
- Three-view Focal Length Recovery From Homographies (CVPR 2025 · similarity 0.82)
- Revisiting Sampson Approximations for Geometric Estimation Problems (CVPR 2024 · similarity 0.81)
- PARSAC: Accelerating Robust Multi-Model Fitting with Parallel Sample Consensus (AAAI 2024 · similarity 0.81)
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