  Registration of images parameterised by landmarks provides a useful method of
describing shape variations by computing the minimum-energy time-dependent
deformation field that flows one landmark set to the other. This is sometimes
known as the geodesic interpolating spline and can be solved via a Hamiltonian
boundary-value problem to give a diffeomorphic registration between images.
However, small changes in the positions of the landmarks can produce large
changes in the resulting diffeomorphism. We formulate a Langevin equation for
looking at small random perturbations of this registration. The Langevin
equation and three computationally convenient approximations are introduced and
used as prior distributions. A Bayesian framework is then used to compute a
posterior distribution for the registration, and also to formulate an average
of multiple sets of landmarks.
