# Consistent and Optimal Solution to Camera Motion Estimation

arXiv: https://arxiv.org/abs/2403.01174 · HTML: https://mlsanity.com/paper/2403.01174/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 57 references that are in the map.
- First posted on arXiv 2 Mar 2024.

## Builds on (most cited first, 15 of 15)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [ORB-SLAM3: An Accurate Open-Source Library for Visual, Visual-Inertial and Multi-Map SLAM](https://mlsanity.com/paper/2007.11898/) (2020)
- [Direct Sparse Odometry](https://mlsanity.com/paper/1607.02565/) (2016)
- [VINS-Mono: A Robust and Versatile Monocular Visual-Inertial State Estimator](https://mlsanity.com/paper/1708.03852/) (2017)
- [Pixel-Perfect Structure-from-Motion with Featuremetric Refinement](https://mlsanity.com/paper/2108.08291/) (2021)
- [Visual Odometry Revisited: What Should Be Learnt?](https://mlsanity.com/paper/1909.09803/) (2019)
- [Deep Keypoint-Based Camera Pose Estimation with Geometric Constraints](https://mlsanity.com/paper/2007.15122/) (2020)
- [An Efficient Solution to Non-Minimal Case Essential Matrix Estimation](https://mlsanity.com/paper/1903.09067/) (2019)
- [MLPnP - A Real-Time Maximum Likelihood Solution to the Perspective-n-Point Problem](https://mlsanity.com/paper/1607.08112/) (2016)
- [Globally Optimal Relative Pose Estimation with Gravity Prior](https://mlsanity.com/paper/2012.00458/) (2020)
- [Five-point Fundamental Matrix Estimation for Uncalibrated Cameras](https://mlsanity.com/paper/1803.00260/) (2018)
- [Certifiable Relative Pose Estimation](https://mlsanity.com/paper/2003.13732/) (2020)
- [Equivalent Constraints for Two-View Geometry: Pose Solution/Pure Rotation Identification and 3D Reconstruction](https://mlsanity.com/paper/1810.05863/) (2018)
- [Global and Asymptotically Efficient Localization from Range Measurements](https://mlsanity.com/paper/2203.16951/) (2022)
- [Consistent and Asymptotically Efficient Localization from Range-Difference Measurements](https://mlsanity.com/paper/2302.03311/) (2023)

## Built on by (most cited first, 1 of 1)

- [Hierarchical Variational Kalman Filtering](https://mlsanity.com/paper/2607.00877/) (2026)

## Abstract (arXiv)

Given 2D point correspondences between an image pair, inferring the camera motion is a fundamental issue in the computer vision community. The existing works generally set out from the epipolar constraint and estimate the essential matrix, which is not optimal in the maximum likelihood (ML) sense. In this paper, we dive into the original measurement model with respect to the rotation matrix and normalized translation vector and formulate the ML problem. We then propose a two-step algorithm to solve it: In the first step, we estimate the variance of measurement noises and devise a consistent estimator based on bias elimination; In the second step, we execute a one-step Gauss-Newton iteration on manifold to refine the consistent estimate. We prove that the proposed estimate owns the same asymptotic statistical properties as the ML estimate: The first is consistency, i.e., the estimate converges to the ground truth as the point number increases; The second is asymptotic efficiency, i.e., the mean squared error of the estimate converges to the theoretical lower bound -- Cramer-Rao bound. In addition, we show that our algorithm has linear time complexity. These appealing characteristics endow our estimator with a great advantage in the case of dense point correspondences. Experiments on both synthetic data and real images demonstrate that when the point number reaches the order of hundreds, our estimator outperforms the state-of-the-art ones in terms of estimation accuracy and CPU time.