Three-view Focal Length Recovery From Homographies

TL;DR

This work tackles focal-length self-calibration from three-view homographies observed on planar scenes. By exploiting the shared plane normal across two homographies, the authors derive explicit constraints that relate the focal lengths to the homographies, enabling efficient solvers for one or two unknown focal lengths. They introduce four solver configurations (Case i–iv) and two solver families, using univariate Sturm-based roots for a single unknown and polynomial eigenvalue methods for two unknowns, achieving substantial speedups and accuracy gains over two-view baselines. A robust LO-RANSAC pipeline integrates these solvers with triplet correspondences (via DLT), culminating in practical focal-length estimation and camera-motion recovery on synthetic and real planar scenes. The paper also provides a public dataset and open-source code to support future research in three-view self-calibration.

Abstract

In this paper, we propose a novel approach for recovering focal lengths from three-view homographies. By examining the consistency of normal vectors between two homographies, we derive new explicit constraints between the focal lengths and homographies using an elimination technique. We demonstrate that three-view homographies provide two additional constraints, enabling the recovery of one or two focal lengths. We discuss four possible cases, including three cameras having an unknown equal focal length, three cameras having two different unknown focal lengths, three cameras where one focal length is known, and the other two cameras have equal or different unknown focal lengths. All the problems can be converted into solving polynomials in one or two unknowns, which can be efficiently solved using Sturm sequence or hidden variable technique. Evaluation using both synthetic and real data shows that the proposed solvers are both faster and more accurate than methods relying on existing two-view solvers. The code and data are available on https://github.com/kocurvik/hf

Figures (7)

• Figure 1: Three cameras view the same plane, defining two homographies $\mathbf{H}_2$, $\mathbf{H}_3$. The two homographies have the same reference image, which should correspond to the same normal vector $\mathbf{n}$.
• Figure 2: Numerical stability of the proposed solvers.
• Figure 3: Focal length errors for the evaluated methods in synthetic experiments. Case i: (a) We vary the proportion of points which lie on the dominant plane with fixed noise $\sigma = 1$. (b, c) We vary noise $\sigma$ with (b) $n_p / n = 1.0$ (i.e. all points lie on a plane) and (c) $n_p / n = 0.5$. Case ii: (d) We vary the proportion of points which lie on the dominant plane with fixed noise $\sigma = 1$. (e) We vary noise $\sigma$ with $n_p / n = 1.0$. (f) We perturb the known focal length $\rho$ so that its error is $\xi_\rho$ with fixed noise $\sigma = 1$ and $n_p / n = 1.0$.
• Figure 4: Median $\xi_f$ and mAA$_f(0.1)$ plotted for different number of RANSAC iterations (10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000). All methods are evaluated in combination with $\mathbf{P}3\mathbf{P}$ding2023revisiting and non-linear optimization within PoseLib poselib. We do not include methods relying solely on pairwise correspondences since they all result in very low accuracy (median $\xi_f>0.4, \text{mAA}_f(0.1)<0.2$).
• Figure 5: The six planar scenes captured in our evaluation dataset.