# Do not Omit Local Minimizer: a Complete Solution for Pose Estimation from 3D Correspondences

## TL;DR

This paper presents a complete, efficient solution for pose estimation from 3D correspondences that accounts for local minima, ambiguous configurations, and is suitable for real-time applications.

## Contribution

It introduces a novel algorithm that reveals local minimizers, solves complex rational equations, and handles ambiguous configurations, advancing pose estimation methods.

## Key findings

- Outperforms previous algorithms with small correspondence sets.
- Achieves same accuracy as global methods when the global minimum is found.
- Suitable for real-time pose estimation applications.

## Abstract

Estimating pose from given 3D correspondences, including point-to-point, point-to-line and point-to-plane correspondences, is a fundamental task in computer vision with many applications. We present a complete solution for this task, including a solution for the minimal problem and the least-squares problem of this task. Previous works mainly focused on finding the global minimizer to address the least-squares problem. However, existing works that show the ability to achieve global minimizer are still unsuitable for real-time applications. Furthermore, as one of contributions of this paper, we prove that there exist ambiguous configurations for any number of lines and planes. These configurations have several solutions in theory, which makes the correct solution may come from a local minimizer. Our algorithm is efficient and able to reveal local minimizers. We employ the Cayley-Gibbs-Rodriguez (CGR) parameterization of the rotation to derive a general rational cost for the three cases of 3D correspondences. The main contribution of this paper is to solve the resulting equation system of the minimal problem and the first-order optimality conditions of the least-squares problem, both of which are of complicated rational forms. The central idea of our algorithm is to introduce intermediate unknowns to simplify the problem. Extensive experimental results show that our algorithm significantly outperforms previous algorithms when the number of correspondences is small. Besides, when the global minimizer is the solution, our algorithm achieves the same accuracy as previous algorithms that have guaranteed global optimality, but our algorithm is applicable to real-time applications.