# Rotation Averaging in a Split Second: A Primal-Dual Method and a Closed-Form for Cycle Graphs

arXiv: https://arxiv.org/abs/2109.08046 · HTML: https://mlsanity.com/paper/2109.08046/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 26 references that are in the map.
- First posted on arXiv 16 Sep 2021.

## Builds on (most cited first, 8 of 8)

- [SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group](https://mlsanity.com/paper/1612.07386/) (2016)
- [A Large Dataset of Object Scans](https://mlsanity.com/paper/1602.02481/) (2016)
- [The non-convex Burer-Monteiro approach works on smooth semidefinite programs](https://mlsanity.com/paper/1606.04970/) (2016)
- [Shonan Rotation Averaging: Global Optimality by Surfing SO(p)n](https://mlsanity.com/paper/2008.02737/) (2020)
- [Lagrangian Duality in 3D SLAM: Verification Techniques and Optimal Solutions](https://mlsanity.com/paper/1506.00746/) (2015)
- [A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints](https://mlsanity.com/paper/1506.00575/) (2015)
- [Parallelizable Algorithms for Optimization Problems with Orthogonality Constraints](https://mlsanity.com/paper/1810.03930/) (2018)
- [GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices](https://mlsanity.com/paper/1812.00426/) (2018)

## Built on by (most cited first, 12 of 12)

- [HARA: A Hierarchical Approach for Robust Rotation Averaging](https://mlsanity.com/paper/2111.08831/) (2021)
- [Advances in Global Solvers for 3D Vision](https://mlsanity.com/paper/2602.14662/) (2026)
- [RAGO: Recurrent Graph Optimizer For Multiple Rotation Averaging](https://mlsanity.com/paper/2212.07211/) (2022)
- [Rotation Synchronization via Deep Matrix Factorization](https://mlsanity.com/paper/2305.05268/) (2023)
- [Towards Initialization-free Calibrated Bundle Adjustment](https://mlsanity.com/paper/2506.23808/) (2025)
- [Certifiably Optimal Anisotropic Rotation Averaging](https://mlsanity.com/paper/2503.07353/) (2025)
- [A Taxonomy of Structure from Motion Methods](https://mlsanity.com/paper/2505.15814/) (2025)
- [Incremental Rotation Averaging Revisited](https://mlsanity.com/paper/2309.16924/) (2023)
- [VICAN: Very Efficient Calibration Algorithm for Large Camera Networks](https://mlsanity.com/paper/2405.10952/) (2024)
- [Rotation Averaging: A Primal-Dual Method and Closed-Forms in Cycle Graphs](https://mlsanity.com/paper/2406.18564/) (2024)
- [A Multi-modal Garden Dataset and Hybrid 3D Dense Reconstruction Framework Based on Panoramic Stereo Images for a Trimming Robot](https://mlsanity.com/paper/2305.06278/) (2023)
- [Making Rotation Averaging Fast and Robust with Anisotropic Coordinate Descent](https://mlsanity.com/paper/2506.01940/) (2025)

## Abstract (arXiv)

A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In spite of being an integral part of bundle adjustment and structure-from-motion, averaging rotations is both a non-convex and high-dimensional optimization problem. In this paper, we address it from a maximum likelihood estimation standpoint and make a twofold contribution. Firstly, we set forth a novel initialization-free primal-dual method which we show empirically to converge to the global optimum. Further, we derive what is to our knowledge, the first optimal closed-form solution for rotation averaging in cycle graphs and contextualize this result within spectral graph theory. Our proposed methods achieve a significant gain both in precision and performance.