# QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition

arXiv: https://arxiv.org/abs/2602.22639 · HTML: https://mlsanity.com/paper/2602.22639/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 56 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Builds on (most cited first, 22 of 22)

- [Graph-Cut RANSAC](https://mlsanity.com/paper/1706.00984/) (2017)
- [SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group](https://mlsanity.com/paper/1612.07386/) (2016)
- [GlueStick: Robust Image Matching by Sticking Points and Lines Together](https://mlsanity.com/paper/2304.02008/) (2023)
- [3D Line Mapping Revisited](https://mlsanity.com/paper/2303.17504/) (2023)
- [Baseline Desensitizing In Translation Averaging](https://mlsanity.com/paper/1901.00643/) (2019)
- [Message Passing Least Squares Framework and its Application to Rotation Synchronization](https://mlsanity.com/paper/2007.13638/) (2020)
- [Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings](https://mlsanity.com/paper/1904.02663/) (2019)
- [A clever elimination strategy for efficient minimal solvers](https://mlsanity.com/paper/1703.05289/) (2017)
- [ShapeFit and ShapeKick for Robust, Scalable Structure from Motion](https://mlsanity.com/paper/1608.02165/) (2016)
- [Minimal Problems for the Calibrated Trifocal Variety](https://mlsanity.com/paper/1611.05947/) (2016)
- [GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices](https://mlsanity.com/paper/1812.00426/) (2018)
- [A New Rank Constraint on Multi-view Fundamental Matrices, and its Application to Camera Location Recovery](https://mlsanity.com/paper/1702.03023/) (2017)
- [A Minimal Closed-Form Solution for Multi-Perspective Pose Estimation using Points and Lines](https://mlsanity.com/paper/1807.09970/) (2018)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [Distortion Varieties](https://mlsanity.com/paper/1610.01860/) (2016)
- [A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion](https://mlsanity.com/paper/2404.11590/) (2024)
- [Averaging Essential and Fundamental Matrices in Collinear Camera Settings](https://mlsanity.com/paper/1912.00254/) (2019)
- [The Quadrifocal Variety](https://mlsanity.com/paper/1501.01266/) (2015)
- [Efficient Detection of Long Consistent Cycles and its Application to Distributed Synchronization](https://mlsanity.com/paper/2407.04260/) (2024)
- [Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor](https://mlsanity.com/paper/2409.09313/) (2024)
- [Cycle-Sync: Robust Global Camera Pose Estimation through Enhanced Cycle-Consistent Synchronization](https://mlsanity.com/paper/2511.02329/) (2025)
- [Higher-Order Group Synchronization](https://mlsanity.com/paper/2505.21932/) (2025)

## Abstract (arXiv)

In structure from motion, quadrifocal tensors capture more information than their pairwise counterparts (essential matrices), yet they have often been thought of as impractical and only of theoretical interest. In this work, we challenge such beliefs by providing a new framework to recover n cameras from the corresponding collection of quadrifocal tensors. We form the block quadrifocal tensor and show that it admits a Tucker decomposition whose factor matrices are the stacked camera matrices, and which thus has a multilinear rank of (4,~4,~4,~4) independent of n. We develop the first synchronization algorithm for quadrifocal tensors, using Tucker decomposition, alternating direction method of multipliers, and iteratively reweighted least squares. We further establish relationships between the block quadrifocal, trifocal, and bifocal tensors, and introduce an algorithm that jointly synchronizes these three entities. Numerical experiments demonstrate the effectiveness of our methods on modern datasets, indicating the potential and importance of using higher-order information in synchronization.