@conference {Berger-et-al-2015a,
	title = {Second Order Minimum Energy Filtering on SE(3) with Nonlinear Measurement Equations},
	booktitle = {Scale Space and Variational Methods in Computer Vision (SSVM 2015)},
	year = {2015},
	publisher = {Springer International Publishing},
	organization = {Springer International Publishing},
	abstract = {Accurate camera motion estimation is a fundamental build- ing block for many Computer Vision algorithms. For improved robustness, temporal consistency of translational and rotational camera velocity is often assumed by propagating motion information forward using stochastic filters. Classical stochastic filters, however, use linear approximations for the non-linear observer model and for the non-linear structure of the underlying Lie Group SE(3) and have to approximate the unknown posteriori distribution. In this paper we employ a non-linear measurement model for the camera motion estimation problem that incorporates multiple observation equations. We solve the underlying filtering problem using a novel Minimum Energy Filter on SE(3) and give explicit expressions for the optimal state variables. Experiments on the challenging KITTI benchmark show that, although a simple motion model is only employed, our approach improves rotational velocity esti- mation and otherwise is on par with the state-of-the-art.},
	doi = {10.1007/978-3-319-18461-6_32},
	url = {http://dx.doi.org/10.1007/978-3-319-18461-6_32},
	author = {Johannes Berger and Andreas Neufeld and Florian Becker and Frank Lenzen and Christoph Schn{\"o}rr}
}
@booklet {Berger-et-al-2015b,
	title = {Second-Order Recursive Filtering on the Rigid-Motion Lie Group SE(3) Based on Nonlinear Observations},
	year = {2015},
	abstract = {Camera motion estimation from observed scene features is an important task in image processing to increase the accuracy of many methods, e.g. optical flow and structure-from-motion. Due to the curved geometry of the state space SE(3) and the non-linear relation to the observed optical flow, many recent filtering approaches use a first-order approximation and assume a Gaussian a posteriori distribution or restrict the state to Euclidean geometry. The physical model is usually also limited to uniform motions. 
We propose a second-order minimum energy filter with a generalized kinematic model that copes with the full geometry of SE(3) as well as with the nonlinear dependencies between the state space and observations. The derived filter enables reconstructing motions correctly for synthetic and real scenes, e.g. from the KITTI benchmark. Our experiments confirm that the derived minimum energy filter with higher-order state differential equation copes with higher-order kinematics and is also able to minimize model noise. We also show that the proposed filter is superior to state-of-the-art extended Kalman filters on Lie groups in the case of linear observations and that our method reaches the accuracy of modern visual odometry methods.},
	keywords = {Constant Acceleration Model, Lie Group, Minimum Energy Filter, Optimal Control, Recursive Filtering, Visual Odometry},
	url = {http://arxiv.org/abs/1507.06810},
	author = {Johannes Berger and Frank Lenzen and Florian Becker and Andreas Neufeld and Christoph Schn{\"o}rr}
}
