About: Riemannian metric and Lie bracket in computational anatomy

Computational anatomy (CA) is the study of shape and form in medical imaging. The study of deformable shapes in computational anatomy rely on high-dimensional diffeomorphism groups which generate orbits of the form . In CA, this orbit is in general considered a smooth Riemannian manifoldsince at every point of the manifold there is an inner product inducing the norm on the tangent spacethat varies smoothly from point to point in the manifold of shapes . This is generated by viewing thegroup of diffeomorphisms as a Riemannian manifold with , associated to the tangent space at . This induces the norm and metric on the orbit under the action from the group of diffeomorphisms.