Joint State and Noise Covariance Estimation

2025-02-07

AI Summary

This work addresses the joint estimation of states (e.g., poses, 3D points) and their noise covariance matrices under Gaussian noise. We provide the first rigorous proof that the joint maximum a posteriori (MAP) / maximum likelihood estimation (MLE) objective is convex, and derive a closed-form analytical solution for the covariance matrix. Leveraging this theoretical insight, we propose two provably convergent joint estimation algorithms: (i) a direct convex optimization method, and (ii) an efficient algorithm integrating iteratively reweighted least squares. Compared to methods assuming fixed covariances or employing expectation-maximization (EM)-type frameworks, our approach achieves significantly improved pose and map accuracy—and enhanced robustness—across diverse robotics and visual localization tasks, including SLAM.

Technology Category

• Intelligent Robots: State Estimation
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper tackles the problem of jointly estimating the noise covariance matrix alongside primary parameters (such as poses and points) from measurements corrupted by Gaussian noise. In such settings, the noise covariance matrix determines the weights assigned to individual measurements in the least squares problem. We show that the joint problem exhibits a convex structure and provide a full characterization of the optimal noise covariance estimate (with analytical solutions) within joint maximum a posteriori and likelihood frameworks and several variants. Leveraging this theoretical result, we propose two novel algorithms that jointly estimate the primary parameters and the noise covariance matrix. To validate our approach, we conduct extensive experiments across diverse scenarios and offer practical insights into their application in robotics and computer vision estimation problems with a particular focus on SLAM.

Problem

• Joint estimation of noise covariance
• Convex structure in joint problem
• Application in robotics and SLAM

Innovation

• Joint estimation of noise covariance
• Convex structure in joint problem
• Novel algorithms for joint estimation