# E-Graph: Minimal Solution for Rigid Rotation with Extensibility Graphs

arXiv: https://arxiv.org/abs/2207.10008 · HTML: https://mlsanity.com/paper/2207.10008/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 20 Jul 2022.

## Builds on (most cited first, 12 of 12)

- [ORB-SLAM: a Versatile and Accurate Monocular SLAM System](https://mlsanity.com/paper/1502.00956/) (2015)
- [ORB-SLAM3: An Accurate Open-Source Library for Visual, Visual-Inertial and Multi-Map SLAM](https://mlsanity.com/paper/2007.11898/) (2020)
- [ORB-SLAM2: an Open-Source SLAM System for Monocular, Stereo and RGB-D Cameras](https://mlsanity.com/paper/1610.06475/) (2016)
- [PL-SLAM: a Stereo SLAM System through the Combination of Points and Line Segments](https://mlsanity.com/paper/1705.09479/) (2017)
- [LDSO: Direct Sparse Odometry with Loop Closure](https://mlsanity.com/paper/1808.01111/) (2018)
- [Stereo DSO: Large-Scale Direct Sparse Visual Odometry with Stereo Cameras](https://mlsanity.com/paper/1708.07878/) (2017)
- [Structure-SLAM: Low-Drift Monocular SLAM in Indoor Environments](https://mlsanity.com/paper/2008.01963/) (2020)
- [RGB-D SLAM with Structural Regularities](https://mlsanity.com/paper/2010.07997/) (2020)
- [ManhattanSLAM: Robust Planar Tracking and Mapping Leveraging Mixture of Manhattan Frames](https://mlsanity.com/paper/2103.15068/) (2021)
- [Minimal Solutions for Relative Pose with a Single Affine Correspondence](https://mlsanity.com/paper/1912.10776/) (2019)
- [Incremental Visual-Inertial 3D Mesh Generation with Structural Regularities](https://mlsanity.com/paper/1903.01067/) (2019)
- [Co-Planar Parametrization for Stereo-SLAM and Visual-Inertial Odometry](https://mlsanity.com/paper/2009.12662/) (2020)

## Built on by (most cited first, 4 of 4)

- [3DRealCar: An In-the-wild RGB-D Car Dataset with 360-degree Views](https://mlsanity.com/paper/2406.04875/) (2024)
- [GIVEPose: Gradual Intra-class Variation Elimination for RGB-based Category-Level Object Pose Estimation](https://mlsanity.com/paper/2503.15110/) (2025)
- [HiNeuS: High-fidelity Neural Surface Mitigating Low-texture and Reflective Ambiguity](https://mlsanity.com/paper/2506.23854/) (2025)
- [U-ARE-ME: Uncertainty-Aware Rotation Estimation in Manhattan Environments](https://mlsanity.com/paper/2403.15583/) (2024)

## Abstract (arXiv)

Minimal solutions for relative rotation and translation estimation tasks have been explored in different scenarios, typically relying on the so-called co-visibility graph. However, how to build direct rotation relationships between two frames without overlap is still an open topic, which, if solved, could greatly improve the accuracy of visual odometry. In this paper, a new minimal solution is proposed to solve relative rotation estimation between two images without overlapping areas by exploiting a new graph structure, which we call Extensibility Graph (E-Graph). Differently from a co-visibility graph, high-level landmarks, including vanishing directions and plane normals, are stored in our E-Graph, which are geometrically extensible. Based on E-Graph, the rotation estimation problem becomes simpler and more elegant, as it can deal with pure rotational motion and requires fewer assumptions, e.g. Manhattan/Atlanta World, planar/vertical motion. Finally, we embed our rotation estimation strategy into a complete camera tracking and mapping system which obtains 6-DoF camera poses and a dense 3D mesh model. Extensive experiments on public benchmarks demonstrate that the proposed method achieves state-of-the-art tracking performance.