# Averaging Essential and Fundamental Matrices in Collinear Camera Settings

arXiv: https://arxiv.org/abs/1912.00254 · HTML: https://mlsanity.com/paper/1912.00254/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2019.

## Built on by (most cited first, 5 of 5)

- [Deep Permutation Equivariant Structure from Motion](https://mlsanity.com/paper/2104.06703/) (2021)
- [Minimal Perspective Autocalibration](https://mlsanity.com/paper/2405.05605/) (2024)
- [On the Robustness of Multi-View Rotation Averaging](https://mlsanity.com/paper/2102.05454/) (2021)
- [QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition](https://mlsanity.com/paper/2602.22639/) (2026)
- [Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices](https://mlsanity.com/paper/2602.23450/) (2026)

## Abstract (arXiv)

Global methods to Structure from Motion have gained popularity in recent years. A significant drawback of global methods is their sensitivity to collinear camera settings. In this paper, we introduce an analysis and algorithms for averaging bifocal tensors (essential or fundamental matrices) when either subsets or all of the camera centers are collinear. We provide a complete spectral characterization of bifocal tensors in collinear scenarios and further propose two averaging algorithms. The first algorithm uses rank constrained minimization to recover camera matrices in fully collinear settings. The second algorithm enriches the set of possibly mixed collinear and non-collinear cameras with additional, "virtual cameras," which are placed in general position, enabling the application of existing averaging methods to the enriched set of bifocal tensors. Our algorithms are shown to achieve state of the art results on various benchmarks that include autonomous car datasets and unordered image collections in both calibrated and unclibrated settings.