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

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 mathbbD mathbbH1. 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 SOleft(3right) of rotation matrices or the hypersphere mathbbH1 of quaternions, we treat the orientation and translation jointly on the dual quaternion quadric in the 7-dimensional real projective space ℝmathbbP7. 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.