Seminar Details
2014-11-25 (14.00) : A Discrete Geodesic Calculus for Shape Space Applications
At Euler building (A002)
Organized by Mathematical Engineering
Speaker :
Wirth, Benedikt
Abstract :
The notion of shapes can be useful in imaging and computer vision applications, for instance in the context of prior information in form of shapes. Unfortunately, shapes are often difficult to deal with since the space of shapes usually has no linear structure. However, the space of shapes can frequently be equipped with a Riemannian metric, turning it into a Riemannian manifold, a structure that can be exploited computationally. We show how a corresponding set of computational tools on a (possibly infinite-dimensional)
Riemannian manifold, a discrete geodesic calculus, can be developed just based on a computationally cheap approximate distance measure. It starts from an energy for discrete paths on the manifold, which gives rise to a notion of discrete geodesics as well as a discrete logarithmic map, a variational definition of a discrete exponential map, and a discrete parallel transport. We investigate the fundamental properties of those discrete operations and their convergence against their continuous counterparts, and we present applications in a space of viscous fluid-like shapes. (Joint work with Martin Rumpf, University of Bonn)