# The Probabilistic Normal Epipolar Constraint for Frame-To-Frame Rotation Optimization under Uncertain Feature Positions

arXiv: https://arxiv.org/abs/2204.02256 · HTML: https://mlsanity.com/paper/2204.02256/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 72 references that are in the map.
- First posted on arXiv 5 Apr 2022.

## Builds on (most cited first, 13 of 13)

- [ORB-SLAM: a Versatile and Accurate Monocular SLAM System](https://mlsanity.com/paper/1502.00956/) (2015)
- [ORB-SLAM2: an Open-Source SLAM System for Monocular, Stereo and RGB-D Cameras](https://mlsanity.com/paper/1610.06475/) (2016)
- [Direct Sparse Odometry](https://mlsanity.com/paper/1607.02565/) (2016)
- [Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age](https://mlsanity.com/paper/1606.05830/) (2016)
- [D3VO: Deep Depth, Deep Pose and Deep Uncertainty for Monocular Visual Odometry](https://mlsanity.com/paper/2003.01060/) (2020)
- [Deep Virtual Stereo Odometry: Leveraging Deep Depth Prediction for Monocular Direct Sparse Odometry](https://mlsanity.com/paper/1807.02570/) (2018)
- [Visual-Inertial Mapping with Non-Linear Factor Recovery](https://mlsanity.com/paper/1904.06504/) (2019)
- [Rotation-Only Bundle Adjustment](https://mlsanity.com/paper/2011.11724/) (2020)
- [Recalibrating the KITTI Dataset Camera Setup for Improved Odometry Accuracy](https://mlsanity.com/paper/2109.03462/) (2021)
- [Monocular Rotational Odometry with Incremental Rotation Averaging and Loop Closure](https://mlsanity.com/paper/2010.01872/) (2020)
- [Geometric Interpretations of the Normalized Epipolar Error](https://mlsanity.com/paper/2008.01254/) (2020)
- [Quaternion Based Camera Pose Estimation From Matched Feature Points](https://mlsanity.com/paper/1704.02672/) (2017)
- [Instant Visual Odometry Initialization for Mobile AR](https://mlsanity.com/paper/2107.14659/) (2021)

## Built on by (most cited first, 8 of 8)

- [Learning Correspondence Uncertainty via Differentiable Nonlinear Least Squares](https://mlsanity.com/paper/2305.09527/) (2023)
- [Full-DoF Egomotion Estimation for Event Cameras Using Geometric Solvers](https://mlsanity.com/paper/2503.03307/) (2025)
- [From Correspondences to Pose: Non-minimal Certifiably Optimal Relative Pose without Disambiguation](https://mlsanity.com/paper/2312.05995/) (2023)
- [DAC: Detector-Agnostic Spatial Covariances for Deep Local Features](https://mlsanity.com/paper/2305.12250/) (2023)
- [An Accurate and Real-time Relative Pose Estimation from Triple Point-line Images by Decoupling Rotation and Translation](https://mlsanity.com/paper/2403.11639/) (2024)
- [VOCA: Visual Odometry with Codec Awareness](https://mlsanity.com/paper/2607.00189/) (2026)
- [DOGE: An Extrinsic Orientation and Gyroscope Bias Estimation for Visual-Inertial Odometry Initialization](https://mlsanity.com/paper/2412.08135/) (2024)
- [IUP-Pose: Decoupled Iterative Uncertainty Propagation for Real-time Relative Pose Regression via Implicit Dense Alignment v1](https://mlsanity.com/paper/2603.19625/) (2026)

## Abstract (arXiv)

The estimation of the relative pose of two camera views is a fundamental problem in computer vision. Kneip et al. proposed to solve this problem by introducing the normal epipolar constraint (NEC). However, their approach does not take into account uncertainties, so that the accuracy of the estimated relative pose is highly dependent on accurate feature positions in the target frame. In this work, we introduce the probabilistic normal epipolar constraint (PNEC) that overcomes this limitation by accounting for anisotropic and inhomogeneous uncertainties in the feature positions. To this end, we propose a novel objective function, along with an efficient optimization scheme that effectively minimizes our objective while maintaining real-time performance. In experiments on synthetic data, we demonstrate that the novel PNEC yields more accurate rotation estimates than the original NEC and several popular relative rotation estimation algorithms. Furthermore, we integrate the proposed method into a state-of-the-art monocular rotation-only odometry system and achieve consistently improved results for the real-world KITTI dataset.