Abstract |
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[eng] In signal processing, we have often to deal with data that belong to non-linear spaces, or manifolds. We can think about the rotation group in pose estimation, the diffusion tensors in image processing, or the Grassmann manifold in subspace tracking techniques. The goal of this thesis is to design algorithms to analyze this kind of data.
We first consider averaging problems on Riemannian manifolds. In particular, we study gradient and Newton methods to compute the Karcher mean of rotation matrices and symmetric positive definite matrices.
Then, we design a gradient method to compute a geodesic that best fits a set of time-labeled data points. We describe the approach on different symmetric spaces of interest in practical applications.
Finally, on some Riemannian manifolds, there is no known algorithm to compute the Riemannian distance between two points. This is the case for the Stiefel manifold, the general linear group, or the symplectic group. We address these problems by developing some optimization schemes that enable us to compute this distance in some cases. |