Second Order Minimum Energy Filtering on $${\text {SE}}_{3}$$ with Nonlinear Measurement Equations

Abstract

Accurate camera motion estimation is a fundamental building 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 SE3 and have to approximate the unknown posteriori distribution. In this paper we employ a nonlinear 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 SE3 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 estimation and otherwise is on par with the state-of-the-art.

Joint Epipolar Tracking (JET): Simultaneous Optimization of Epipolar Geometry and Feature Correspondences

“…For instance, the authors of [2,23,25,27,28] use epipolar constraints for stabilizing discrete matching, whereas Valgaerts et al [26] proposed a variational approach to estimate dense optical flow and the epipolar geometry (represented by the fundamental matrix) simultaneously. Other direct methods that also explicitly take into consideration the depth structure of the scene are [4,18,20].…”

Attitude estimation with feedback particle filter

“…In this section, a Galerkin scheme is presented to approximate the solution of the Poisson equation (9). Since the equations for each j = 1, 2, ..., m are uncoupled, without loss of generality, a scalar-valued measurement is assumed (i.e., m = 1, and φ j , h j are denoted as φ , h).…”

“…In a kernel-based scheme, the solution to the Poisson equation (9) is the solution of the following fixed-point equation for fixed positive τ,…”

“…A class of symmetry-preserving observers have been proposed to exploit certain invariance properties [11], [12], leading to the invariant EKF algorithm [13], [6], [3], the invariant ensemble EKF [6], and the invariant particle filter [5] within the stochastic filtering framework. Filters based on certain variational formulations on Lie groups have also been investigated [44], [9], [26]. Particle filters for attitude Financial support from the NSF CMMI grants 1334987 and 1462773 is gratefully acknowledged.…”

Joint Recursive Monocular Filtering of Camera Motion and Disparity Map

“…Zamani et al [29] introduces so-called minimum energy filters for linear filtering problems for compact Lie groups based on optimal control theory and the recursive filtering principle of Mortensen [18]. This approach was generalized to (non-)compact Lie groups in [25] and applied to a non-linear filtering problem on SE 3 for camera motion estimation [4].…”