# Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor

arXiv: https://arxiv.org/abs/2409.09313 · HTML: https://mlsanity.com/paper/2409.09313/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 57 references that are in the map.
- First posted on arXiv 14 Sep 2024.

## Builds on (most cited first, 27 of 27)

- [A Survey of Structure from Motion](https://mlsanity.com/paper/1701.08493/) (2017)
- [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)
- [Robust Camera Location Estimation by Convex Programming](https://mlsanity.com/paper/1412.0165/) (2014)
- [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)
- [PLMP -- Point-Line Minimal Problems in Complete Multi-View Visibility](https://mlsanity.com/paper/1903.10008/) (2019)
- [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)
- [Regularized Tyler's Scatter Estimator: Existence, Uniqueness, and Algorithms](https://mlsanity.com/paper/1407.3079/) (2014)
- [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)
- [Robust Multi-object Matching via Iterative Reweighting of the Graph Connection Laplacian](https://mlsanity.com/paper/2006.06658/) (2020)
- [On the Instability of Relative Pose Estimation and RANSAC's Role](https://mlsanity.com/paper/2112.14651/) (2021)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [Distortion Varieties](https://mlsanity.com/paper/1610.01860/) (2016)
- [Fast, Accurate and Memory-Efficient Partial Permutation Synchronization](https://mlsanity.com/paper/2203.16505/) (2022)
- [A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion](https://mlsanity.com/paper/2404.11590/) (2024)
- [Trifocal Relative Pose from Lines at Points and its Efficient Solution](https://mlsanity.com/paper/1903.09755/) (2019)
- [Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching](https://mlsanity.com/paper/2201.04797/) (2022)
- [Efficient Detection of Long Consistent Cycles and its Application to Distributed Synchronization](https://mlsanity.com/paper/2407.04260/) (2024)
- [Algebraic Constraints and Algorithms for Common Lines in Cryo-EM](https://mlsanity.com/paper/2403.16879/) (2024)

## Built on by (most cited first, 3 of 3)

- [Higher-Order Group Synchronization](https://mlsanity.com/paper/2505.21932/) (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)

## Abstract (arXiv)

The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera poses (locations and orientations up to a global transformation) from the block trifocal tensor. We establish an explicit Tucker factorization of this tensor, revealing a low multilinear rank of (6,4,4) independent of the number of cameras under appropriate scaling conditions. We prove that this rank constraint provides sufficient information for camera recovery in the noiseless case. The constraint motivates a synchronization algorithm based on the higher-order singular value decomposition of the block trifocal tensor. Experimental comparisons with state-of-the-art global synchronization methods on real datasets demonstrate the potential of this algorithm for significantly improving location estimation accuracy. Overall this work suggests that higher-order interactions in synchronization problems can be exploited to improve performance, beyond the usual pairwise-based approaches.