NDLI: Continuous-Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions

Content Provider	Springer Nature Link
Author	Bourmaud, Guillaume Mégret, Rémi Arnaudon, Marc Giremus, Audrey
Copyright Year	2014
Abstract	In this paper we generalize the continuous-discrete extended Kalman filter (CD-EKF) to the case where the state and the observations evolve on connected unimodular matrix Lie groups. We propose a new assumed density filter called continuous-discrete extended Kalman filter on Lie groups (CD-LG-EKF). It is built upon a geometrically meaningful modeling of the concentrated Gaussian distribution on Lie groups. Such a distribution is parametrized by a mean and a covariance matrix defined on the Lie group and in its associated Lie algebra respectively. Our formalism yields tractable equations for both non-linear continuous time propagation and discrete update of the distribution parameters under the assumption that the posterior distribution of the state is a concentrated Gaussian. As a side effect, we contribute to the derivation of the first and second order differential of the matrix Lie group logarithm using left connection. We also show that the CD-LG-EKF reduces to the usual CD-EKF if the state and the observations evolve on Euclidean spaces. Our approach leads to a systematic methodology for the design of filters, which is illustrated by the application to a camera pose filtering problem with observations on Lie group. In this application, the CD-LG-EKF significantly outperforms two constrained non-linear filters (one based on a linearization technique and the other on the unscented transform) applied on the embedding space of the Lie group.
Starting Page	209
Ending Page	228
Page Count	20
File Format	PDF
ISSN	09249907
Journal	Journal of Mathematical Imaging and Vision
Volume Number	51
Issue Number	1
e-ISSN	15737683
Language	English
Publisher	Springer US
Publisher Date	2014-07-10
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Extended Kalman filter Matrix Lie group Filtering on manifold Stochastic processes on manifolds Camera pose filtering Image Processing and Computer Vision Applications of Mathematics Signal, Image and Speech Processing Mathematical Methods in Physics
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology Applied Mathematics Statistics and Probability Condensed Matter Physics Modeling and Simulation Computer Vision and Pattern Recognition

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