# Camera Pose Filtering with Local Regression Geodesics on the Riemannian Manifold of Dual Quaternions

## TL;DR

This paper introduces a novel Riemannian manifold-based local regression filter for smooth, accurate 3D camera pose estimation using dual quaternions, outperforming existing methods in robustness and precision.

## Contribution

It proposes a new manifold-aware filtering method on dual quaternions for joint orientation and translation smoothing, with an outlier-robust IRLS algorithm.

## Key findings

- Improved pose smoothing accuracy on synthetic and real data.
- Enhanced robustness to outliers in pose estimation.
- Demonstrated theoretical and practical advantages over existing methods.

## Abstract

Time-varying, smooth trajectory estimation is of great interest to the vision community for accurate and well behaving 3D systems. In this paper, we propose a novel principal component local regression filter acting directly on the Riemannian manifold of unit dual quaternions $\mathbb{D} \mathbb{H}_1$. We use a numerically stable Lie algebra of the dual quaternions together with $\exp$ and $\log$ operators to locally linearize the 6D pose space. Unlike state of the art path smoothing methods which either operate on $SO\left(3\right)$ of rotation matrices or the hypersphere $\mathbb{H}_1$ of quaternions, we treat the orientation and translation jointly on the dual quaternion quadric in the 7-dimensional real projective space $\mathbb{R}\mathbb{P}^7$. We provide an outlier-robust IRLS algorithm for generic pose filtering exploiting this manifold structure. Besides our theoretical analysis, our experiments on synthetic and real data show the practical advantages of the manifold aware filtering on pose tracking and smoothing.