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**Abstract**

The problem of image registration is the following. We model an object (say human organs or tissues as in the case of medical imaging) as a compact Riemannian manifold without boundary $(M,g)$. Given a template image $I_0:M\to \mathbb R$ of $(M,g)$ of the object and another image $I:M\to \mathbb R$ (perhaps taken at a later time or at a different viewing angle), we seek to find a diffeomorphism $\phi: M\to M$ such that we can match the image to the template: $I= I_0 \circ \phi^{-1}$. In this talk, I will introduce a gradient flow problem whose solution is a diffeomorphism which solves a related image registration problem. I will then discuss the well-posedness of the gradient flow problem. The key innovation of the gradient flow approach to image registration that I will introduce is that the flow takes into account the distortion effects of the diffeomorphism on both the template image $I_0$ and on the object geometry as given by the metric $g$. I will also review a bit of the history of image registration and the large deformation diffeomorphic metric mapping method as it relates to the gradient flow problem I will present.

This is joint work with Klas Modin & Carl-Joar Karlsson at Chalmers University of Technology and the University of Gothenburg.