Three-view Focal Length Recovery From Homographies

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

• The paper presents a novel algebraic method using polynomial constraints from normal vector consistency across three views.
• The method yields faster and more precise focal length recovery than traditional two-view approaches, as verified with both synthetic and real-world data.
• The research provides robust solutions for various camera setups, enabling improved camera calibration for 3D reconstruction and augmented reality applications.

An Analytical Approach to Three-View Focal Length Recovery from Homographies

The paper "Three-view Focal Length Recovery From Homographies" tackles a significant problem in computer vision: estimating focal lengths from three-view homographies through an efficient algebraic approach. This problem has profound implications for camera calibration, particularly in scenarios where intrinsic camera parameters are partially known. The authors introduce novel solutions by capitalizing on the geometric consistency of normal vectors among multiple homographies derived from three camera views.

Methodological Contributions

The research presented leverages the consistency of the normal vectors of planes observed in three different views to derive polynomial constraints between focal lengths and homographies. Depending on the camera setup, four cases are considered:

1. Three cameras with an equal unknown focal length.
2. Three cameras with two different unknown focal lengths.
3. One camera with a known focal length and two others with equal unknown focal lengths.
4. One camera with a known focal length and two others with different unknown focal lengths.

For each case, the authors derive solutions to polynomial equations using modern algebraic techniques such as Sturm sequences and hidden variable methods. This approach significantly streamlines the computational complexity compared to existing two-view solutions, yielding solvers that are faster and demonstrate higher accuracy.

Empirical Evaluation

Both synthetic and real-world datasets were deployed to evaluate the proposed methods, showing that the proposed solvers outperformed two-view solver baselines in terms of speed and precision. The study further introduces a new dataset consisting of six scenes, captured with 14 different cameras, to provide robust benchmarking for focal length recovery methods.

Practical Implications

The practical implications of these findings are noteworthy. Efficiently estimating camera focal lengths from three views enables more accurate camera calibration, which is crucial for applications such as 3D reconstruction and augmented reality. The improvements over traditional two-view methods demonstrate the potential for real-time applications where computational efficiency and accuracy are paramount.

Theoretical Implications and Future Directions

From a theoretical perspective, the paper illustrates the power of combining homography-based approaches with algebraic techniques to tackle complex geometric vision problems. The constraints derived for three-view configurations pave the way for further research into multi-view systems and the automation of these techniques in real-world applications.

Going forward, advancements could focus on extending these methods to non-planar scenes or incorporating them into comprehensive vision systems that require minimal human intervention. This paper lays a vital foundation for such advancements by addressing the critical challenge of focal length estimation using homographies, thereby pushing the boundaries of what can be achieved in camera self-calibration.