Title: "Fitting on manifolds with Bézier functions"
Speaker: Pierre-Yves Gousenbourger
Location: "Shannon" Seminar Room, Place du Levant 3, Maxwell Building, 1st floor
Date / Time (duration): Thursday 16/03/2017, 16h15 (~45')
Abstract: Fitting and interpolation are well known topics on the Euclidean space. However, when it turns out that the data points are manifold valued (understand: when the data point belongs to a certain manifold), most of the algorithms are no more applicable because even the notion of distance is not as simple as on the Euclidean space.
Manifold-valued data appear in various domains as MRI acquisition (diffusion tensors, segmented shapes,...), fluid mechanics (SDP matrices or values on the Grassmann manifold), matrix completion and many more. A particular example concern the representation of local wind fields in a given domain. These can be expressed as covariance matrices which are SDP and of fixed (small) rank. The wind field changes when the prevalent wind has different orientation or magnitude. The computation of this wind field, however, resort in solving complicated and time-consuming PDEs in fluid mechanics. In order to obtain approached solutions as fast as possible, a solution is to compute several wind field covariance matrices representations for a set of given orientation/magnitude of the prevalent wind and then interpolate (or fit) a manifold-valued curve in the space of fixed rand SDP matrices.
In this talk, I will propose a method to fit manifold-valued data points based on a generalization of Bezier curves. This method is motivated by (but not specific to) the wind field orientation problem. I will try to introduce the concepts of manifolds and Bezier curves as a tutorial to finally show interesting results for that specific application.
Last updated December 06, 2012, at 02:25 PM