Málstofa í stærðfræði
Fyrirlesari: Sylvain Arguillère
Titill: Constrained Shape Analysis Through Flows of Diffeomorphisms
Staðsetning: TG-227 (Tæknigarður, 2. hæð)
Tími: Föstudagur 24. júní kl. 13:20.
The general purpose of shape analysis is to compare different shapes in a way that takes into account their geometric properties, such as smoothness, number of self-intersection points, convexity… One way to do this is to find a flow of diffeomorphisms that brings one (template) shape as close as possible to the other (target) shape while minimizing a certain energy. This is the so-called LDDMM method (Large Deformation Diffeomorphic Metric Matching).
Finding this minimizing flow requires solving an optimal control problem that can be seen as looking for (sub-)Riemannian geodesics on the infinite dimensional group of diffeomorphisms with respect to a right-invariant (sub-)Riemannian structure, creating a framework reminiscent of fluid mechanics, and opening the door to some new and exciting infinite dimensional geometries.
In this talk, I will introduce all these concepts, and give the geodesic equations for such structures. Then, I will extend this framework to the case where constraints are added to the shape, in order to better describe the objects they represent, and give some applications in computational anatomy.