# A Minimal Solution for Two-view Focal-length Estimation using Two Affine Correspondences

arXiv: https://arxiv.org/abs/1706.01649 · HTML: https://mlsanity.com/paper/1706.01649/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2017.

## Built on by (most cited first, 14 of 14)

- [DKM: Dense Kernelized Feature Matching for Geometry Estimation](https://mlsanity.com/paper/2202.00667/) (2022)
- [Learning to Solve Hard Minimal Problems](https://mlsanity.com/paper/2112.03424/) (2021)
- [Galois/monodromy groups for decomposing minimal problems in 3D reconstruction](https://mlsanity.com/paper/2105.04460/) (2021)
- [RePoseD: Efficient Relative Pose Estimation With Known Depth Information](https://mlsanity.com/paper/2501.07742/) (2025)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [FastMap: Revisiting Structure from Motion through First-Order Optimization](https://mlsanity.com/paper/2505.04612/) (2025)
- [Relative Pose from SIFT Features](https://mlsanity.com/paper/2203.07930/) (2022)
- [Learning Affine Correspondences by Integrating Geometric Constraints](https://mlsanity.com/paper/2504.04834/) (2025)
- [On Relative Pose Recovery for Multi-Camera Systems](https://mlsanity.com/paper/2102.11996/) (2021)
- [Indoor 3D Reconstruction with an Unknown Camera-Projector Pair](https://mlsanity.com/paper/2407.01945/) (2024)
- [A Minimal Solver for Relative Pose Estimation with Unknown Focal Length from Two Affine Correspondences](https://mlsanity.com/paper/2512.22833/) (2025)
- [PLMP -- Point-Line Minimal Problems for Projective SfM](https://mlsanity.com/paper/2503.04351/) (2025)
- [Affine Correspondences in Stereo Vision: Theory, Practice, and Limitations](https://mlsanity.com/paper/2603.01836/) (2026)
- [Gravity-aware partially calibrated absolute pose estimation from affine- or rotation-covariant features](https://mlsanity.com/paper/2608.20056/) (2026)

## Abstract (arXiv)

A minimal solution using two affine correspondences is presented to estimate the common focal length and the fundamental matrix between two semi-calibrated cameras - known intrinsic parameters except a common focal length. To the best of our knowledge, this problem is unsolved. The proposed approach extends point correspondence-based techniques with linear constraints derived from local affine transformations. The obtained multivariate polynomial system is efficiently solved by the hidden-variable technique. Observing the geometry of local affinities, we introduce novel conditions eliminating invalid roots. To select the best one out of the remaining candidates, a root selection technique is proposed outperforming the recent ones especially in case of high-level noise. The proposed 2-point algorithm is validated on both synthetic data and 104 publicly available real image pairs. A Matlab implementation of the proposed solution is included in the paper.