Estimation de paramètres évoluant sur des groupes de Lie : Application à la cartographie et à la localisation d'une caméra monoculaire

Abstract : In this thesis, we derive novel parameter estimation algorithms dedicated to parameters evolving on Lie groups. These algorithms are casted in a Bayesian formalism, which allows us to establish a notion of uncertainty for the estimated parameters. To do so, a generalization of the multivariate normal distribution to Lie groups, called concentrated normal distribution on Lie groups, is employed. In a first part, we generalize the Continuous-Discrete Extended Kalman Filter (CD-EKF), as well as the Discrete Extended Kalman Filter (D-EKF), to the case where the state and the observations evolve on Lie groups. We obtain two novel algorithms called Continuous-Discrete Extended Kalman Filter on Lie Groups (CD-LG-EKF) and Discrete Extended Kalman Filter on Lie Groups (D-LG-EKF). In a second part, we focus on bridging the gap between the formulation of intrinsic non linear least squares criteria and Kalman filtering/smoothing on Lie groups. We propose a generalization of the Euclidean Iterated Extended Kalman Filter (IEKF) to Lie groups, called LG-IEKF. We also derive a generalization of the Rauch-Tung- Striebel smoother (RTS), also known as Extended Kalman Smoother, to Lie groups, called LG-RTS. Finally, the concepts and algorithms presented in the thesis are employed in a series of applications. Firstly, we propose a novel simultaneous localization and mapping approach. Secondly we develop an indoor camera localization framework. For this latter purpose, we derived a novel Rao-Blackwellized particle smoother on Lie groups, which builds upon the LG-IEKF and the LG-RTS.
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Guillaume Bourmaud. Estimation de paramètres évoluant sur des groupes de Lie : Application à la cartographie et à la localisation d'une caméra monoculaire. Traitement du signal et de l'image [eess.SP]. Université de Bordeaux 2015. Français. ⟨tel-01265293⟩

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