# GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices

arXiv: https://arxiv.org/abs/1812.00426 · HTML: https://mlsanity.com/paper/1812.00426/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2018.

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

- [Global Structure-from-Motion Revisited](https://mlsanity.com/paper/2407.20219/) (2024)
- [Deep Permutation Equivariant Structure from Motion](https://mlsanity.com/paper/2104.06703/) (2021)
- [Rotation Averaging in a Split Second: A Primal-Dual Method and a Closed-Form for Cycle Graphs](https://mlsanity.com/paper/2109.08046/) (2021)
- [Learning Structure-from-Motion with Graph Attention Networks](https://mlsanity.com/paper/2308.15984/) (2023)
- [A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion](https://mlsanity.com/paper/2404.11590/) (2024)
- [Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor](https://mlsanity.com/paper/2409.09313/) (2024)
- [A Taxonomy of Structure from Motion Methods](https://mlsanity.com/paper/2505.15814/) (2025)
- [PRaDA: Projective Radial Distortion Averaging](https://mlsanity.com/paper/2504.16499/) (2025)
- [Higher-Order Group Synchronization](https://mlsanity.com/paper/2505.21932/) (2025)
- [On the Robustness of Multi-View Rotation Averaging](https://mlsanity.com/paper/2102.05454/) (2021)
- [An Algebraic Geometry Approach to Viewing Graph Solvability](https://mlsanity.com/paper/2504.03637/) (2025)
- [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)
- [Learning Global Camera Poses from Noisy View-Graphs for Structure from Motion](https://mlsanity.com/paper/2609.09491/) (2026)

## Abstract (arXiv)

This paper addresses the problem of recovering projective camera matrices from collections of fundamental matrices in multiview settings. We make two main contributions. First, given n choose 2 fundamental matrices computed for n images, we provide a complete algebraic characterization in the form of conditions that are both necessary and sufficient to enabling the recovery of camera matrices. These conditions are based on arranging the fundamental matrices as blocks in a single matrix, called the n-view fundamental matrix, and characterizing this matrix in terms of the signs of its eigenvalues and rank structures. Secondly, we propose a concrete algorithm for projective structure-from-motion that utilizes this characterization. Given a complete or partial collection of measured fundamental matrices, our method seeks camera matrices that minimize a global algebraic error for the measured fundamental matrices. In contrast to existing methods, our optimization, without any initialization, produces a consistent set of fundamental matrices that corresponds to a unique set of cameras (up to a choice of projective frame). Our experiments indicate that our method achieves state of the art performance in both accuracy and running time.